popSimManu.tex

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\newcommand{\Stdpopsim}{\texttt{Stdpopsim}\xspace}
\newcommand{\stdpopsim}{\texttt{stdpopsim}\xspace}
\newcommand{\dadi}{$\partial a \partial i$\xspace}
\newcommand{\MSMC}{\texttt{MSMC}\xspace}
\newcommand{\smcpp}{\texttt{smc++}\xspace}
\newcommand{\stairwayplot}{\texttt{stairway plot}\xspace}
\newcommand{\fastsimcoal}{\texttt{fastsimcoal2}\xspace}
\newcommand{\tskit}{\texttt{tskit}\xspace}

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\rhead{\texttt{stdpopsim}}
\lhead{PopSim Consortium}
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\begin{document}

\title{A community-maintained standard library of population genetic models}
% First authors
\author[1,$\star$]{Jeffrey R. Adrion}
\author[2,$\star$]{Christopher B. Cole}
\author[3,$\star$]{Noah Dukler}
\author[1,$\star$]{Jared G. Galloway}
\author[4,$\star$]{Ariella L. Gladstein}
\author[5,$\star$]{Graham Gower}
\author[6,$\star$]{Christopher C. Kyriazis}
\author[7,$\star$]{Aaron P. Ragsdale}
\author[8,$\star$]{Georgia Tsambos}
% Second Authors
\author[9]{Franz Baumdicker}
\author[10]{Jedidiah Carlson}
\author[11]{Reed A. Cartwright}
\author[12]{Arun Durvasula}
\author[13]{Ilan Gronau}
\author[14]{Bernard Y. Kim}
\author[15]{Patrick McKenzie}
\author[16]{Philipp W. Messer}
\author[17]{Ekaterina Noskova}
\author[18]{Diego Ortega-Del Vecchyo}
\author[5]{Fernando Racimo}
\author[19]{Travis J. Struck}
%Senior Authors
\author[7,$\dagger$]{Simon Gravel}
\author[19,$\dagger$]{Ryan N. Gutenkunst}
\author[6,$\dagger$]{Kirk E. Lohmueller}
\author[1,20,$\dagger$]{Peter L. Ralph}
\author[4,$\dagger$]{Daniel R. Schrider}
\author[3,$\dagger$]{Adam Siepel}
\author[21,$\dagger$,$\aleph$]{Jerome Kelleher}
\author[1,$\dagger$,$\aleph$]{Andrew D. Kern}

\affil[1]{Department of Biology and Institute of Ecology and Evolution, University of Oregon}
\affil[2]{Wellcome Trust Centre for Human Genetics, University of Oxford}
\affil[3]{Simons Center for Quantitative Biology, Cold Spring Harbor Laboratory}
\affil[4]{Department of Genetics, University of North Carolina at Chapel Hill}
\affil[5]{Lundbeck GeoGenetics Centre, Globe Institute, University of Copenhagen}
\affil[6]{Department of Ecology and Evolutionary Biology, University of California, Los Angeles}

\affil[7]{Department of Human Genetics, McGill University}
\affil[8]{Melbourne Integrative Genomics, School of Mathematics and Statistics, University of Melbourne}
\affil[9]{Department of Mathematical Stochastics, University of Freiburg}
\affil[10]{Department of Genome Sciences, University of Washington}
\affil[11]{The Biodesign Institute and The School of Life Sciences, Arizona State University}
\affil[12]{Department of Human Genetics, David Geffen School of Medicine, University of California, Los Angeles}
\affil[13]{The Efi Arazi School of Computer Science, The Herzliya Interdisciplinary Center, Israel}
\affil[14]{Department of Biology, Stanford University}
\affil[15]{Department of Ecology, Evolution, and Environmental Biology, Columbia University}
\affil[16]{Department of Computational Biology, Cornell University}
\affil[17]{Computer Technologies Laboratory, ITMO University}
\affil[18]{International Laboratory for Human Genome Research, National Autonomous University of Mexico}
\affil[19]{Department of Molecular and Cellular Biology, University of Arizona}
\affil[20]{Department of Mathematics, University of Oregon}
\affil[21]{Big Data Institute, Li Ka Shing Centre for Health Information and
Discovery, University of Oxford}
\affil[$\star$]{Denotes shared first authorship, listed alphabetically}
\affil[$\dagger$]{Denotes shared senior authorship, listed alphabetically}
\affil[$\aleph$]{Denotes corresponding authors, listed alphabetically}

\abstract{
The explosion in population genomic data demands ever more complex
modes of analysis, and increasingly these analyses depend on sophisticated simulations.
Recent advances in population genetic simulation have made it possible to
simulate large and complex models, but specifying such models for a particular
simulation engine remains a difficult and error-prone task.
Computational genetics researchers currently re-implement simulation models independently,
leading to inconsistency and duplication of effort.
This situation presents a major barrier to empirical researchers seeking
to use simulations for power analyses of upcoming studies
or sanity checks on existing genomic data.
Population genetics, as a field,
also lacks standard benchmarks by which new tools for inference might be measured.
Here we describe a new resource, \stdpopsim, that attempts to rectify this situation.
\Stdpopsim is a community-driven open source project, which provides easy access to a growing
catalog of published simulation models from a range of organisms
and supports multiple simulation engine backends.
This resource is available as a well-documented python library
with a simple command-line interface.
We share some examples demonstrating how \stdpopsim
can be used to systematically compare demographic inference methods,
and we encourage a broader
community of developers to contribute to this growing resource.
}
\date{}

\keywords{Population genetics, Simulation, Inference, Reproducibility}

\maketitle


\section*{Introduction}
While population genetics has always used statistical methods to make inferences from data,
the degree of sophistication of the questions, models, data, and computational approaches
used have all increased over the past two decades. Currently there exist a myriad of computational methods
that can infer the histories of populations
\citep{gutenkunst2009inferring,li2011inference,excoffier2013robust,schiffels2014inferring, terhorst2017robust,ragsdale2019models},
the distribution of fitness effects \citep{Boyko:2008cr,kim2017inference,tataru2017inference,Fortier703918,Huang2019,Vecchyo770966},
recombination rates \citep{mcvean2004ldhat,chan2012genome,lin2013fast,adrion2020predicting,Barroso2019},
and the extent of positive selection in genome sequence data
\citep{kim2002detecting,eyre2009estimating, alachiotis2012omegaplus,garud2015sweeps,degiorgio2016sweepfinder2,kern2018diplos,sugden2018localization}.
While these methods have undoubtedly increased our understanding of
genetic and evolutionary processes, very little has been done to systematically
benchmark the quality of these inferences or their robustness to deviations from their underlying assumptions.
As large databases of population genetic variation begin to be used to inform public health procedures,
the accuracy and quality of these inferences is becoming ever more important.

Assessing the accuracy of inference methods for population genetics is
challenging in large part because the ``ground-truth'' in question
generally comes not from direct empirical observations, as the relevant
historical processes can rarely be observed, but instead from simulations.
Population genetic simulations are therefore critically important to the
field, yet there has been no systematic attempt to establish community
standards or best practices for executing them.
Instead, the general modus
operandi to date has been for individual groups to validate their own
methods using simulations coded from scratch.
%ACS: this level of detail seems a bit superfluous here
%(perhaps provided as long command-line calls in supplementary material).
Often these simulations are more useful to showcase a novel method than to
rigorously compare it with competing methods.
Moreover, this situation results in a great deal of duplicated effort,
and contributes to decreased reproducibility and transparency across the entire field.
It is also a barrier to entry to the field, because new researchers can
struggle with the many steps involved
in implementing a state-of-the-art population genetics simulation,
including identifying appropriate demographic models from the literature,
translating them into input for a simulator, and choosing appropriate values for key population genetic parameters,
such as the mutation and recombination rates.

%ACS: This paragraph seems to me to need a more explicit connection to
%the enterprise of providing standardized simulations.  It feels a bit tangential
%as written.  I have tried to edit it accordingly.
A related issue is that it has been challenging to assess the degree to which modeling assumptions
and choices of data summaries can affect population genetic inferences.
Standardized simulations would enable these questions to be systematically examined.
Importantly, there are clear examples of different methods yielding fundamentally
different conclusions. For example, Markovian coalescent methods applied to human genomes have
suggested large ancient ($>100,000$ years ago) ancestral population sizes and
bottlenecks that have not been detected by other methods based on allele frequency spectra
\citep[see][]{beichman2017comparison}.
These distinct methods differ in how they model, summarize, and optimize fit to
genetic variation data, suggesting that such design choices can greatly affect the
performance of the inference. Furthermore, some methods are likely to
perform better than others under certain scenarios, but
researchers lack principled guidelines for selecting the best method for addressing
their particular questions. The need for
guidance from simulated data will only increase as researchers
seek to apply population genetic methods to a growing collection of non-model taxa.

For these reasons, we have generated a standardized, community-driven resource
for simulating published demographic models from a number of popular study systems.
This resource, which we call \stdpopsim, makes running
realistic simulations for population genetic analysis a simple matter of
choosing pre-implemented models from a community-maintained catalog.
The \stdpopsim catalog currently contains five species: humans,
\textit{Drosophila melanogaster}, \textit{Arabidopsis thaliana},
\textit{Escherichia coli}, and \textit{Canis familiaris}.
For each species, the catalog contains information on our current understanding of
the physical organization of its genome (e.g., chromosome structure),
one or more inferred genetic maps (except for \textit{E.~coli}),
and default population-level parameters (mutation rate and generation time estimates).
The catalog also contains numerous published demographic models for the first three species.
These available models and parameters are meant to represent the field's current understanding,
and we intend for this resource to evolve as new results become available, and relevant
existing models are added to \stdpopsim by the community.
We have implemented both a command line interface and a simple Python API
that can be used to simulate genomic data
from a choice of organism, genetic map, chromosome, and demographic history.
In this way, \stdpopsim will lower the barrier to high-quality simulation for exploratory analyses,
enable rigorous evaluation of population genetic software,
and contribute to increased reliability of population genetic inferences.


The \stdpopsim library has been developed by the PopSim Consortium using a
distributed open source model, with strong procedures in place
to continue its growth and maintain its quality.
Importantly, we developed rigorous quality control methods to ensure that we have
correctly implemented the models as described in their original publication
and provided documented methods for others to contribute new models.
We invite new collaborators to join our community:
those interested should visit our developer documentation at
\url{https://stdpopsim.readthedocs.io/en/latest/development.html}.
Below we describe the resource and give
examples of how it can be used to benchmark demographic inference methods.





\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\linewidth]{display_items/Figure1.pdf}
    \caption{\textbf{Structure of \stdpopsim}. \textbf{(A)} The
hierarchical organization of the \stdpopsim catalog contains all model simulation information
within individual species (expanded information shown here for \textit{H.~sapiens} only).
Each species is associated with a representation of the physical genome, and one or more genetic maps and demographic models.
Dotted lines indicate that only a subset of these categories is shown.
At right we show example code to specify
and simulate models using \textbf{(B)} the python API or \textbf{(C)} the command line interface.
    }
\label{fig:cartoon}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Results}

%ACS: the consortium really isn't the point; let's get right to describing what stdpopsim is
%The first contribution of the PopSim consortium is
The \stdpopsim library is a
a community-maintained collection of empirical genome data and population genetics simulation models,
with a structure depicted in Figure \ref{fig:cartoon}.
The package centers on a catalog of genomic information and demographic models
for a growing list of species (Fig.~\ref{fig:cartoon}A),
and software resources to facilitate efficient simulations (Fig.~\ref{fig:cartoon}B-C).

Given the genome data and simulation model descriptions defined within the
library, it is straightforward to run standardized simulations
across a range of organisms. \Stdpopsim has a Python API and a user-friendly
command line interface, allowing users with minimal experience direct access to
state-of-the-art simulations. Simulations are output in the ``tree sequence''
format \citep{kelleher2016efficient,kelleher2018efficient,kelleher2019inferring}, which
contains complete genealogical information about the simulated samples, is
extremely compact, and can be processed efficiently using the \tskit library
\citep{kelleher2016efficient,kelleher2018efficient}.
The tree sequence format could also be converted
to other formats (e.g., VCF) by the user if desired.

\subsection*{The species catalog}

The central feature of \stdpopsim is the species catalog, a systematic organization
of the key quantitative data needed to simulate a given species.
Data is currently available for humans,
\textit{D.~melanogaster}, \textit{A.~thaliana},
\textit{Escherichia coli}, and \textit{Canis familiaris}.
A species definition consists of two key elements.
Firstly, the library defines some basic information about our current understanding of each
species' genome, including information about chromosome
lengths, average mutation rate estimates, and generation times.
We also provide access to detailed empirical information such as inferred genetic maps,
which model observed heterogeneity in recombination rate along chromosomes.
Such maps are often large,
so we do not distribute them directly with the software, but make them available
for download in a standard format.
When a simulation using such a map is requested by the user,
\stdpopsim will transparently download the map data into a local cache,
% (\texttt{$\sim$/.local/cache/stdpopsim} by default on Unix platforms)
where it can be quickly retrieved for subsequent simulations.
In the initial version of \stdpopsim we support
the HapMapII~\citep{international2007second} and
deCODE~\citep{kong2010fine} genetic maps for humans;
the \cite{salome2011recombination} map for \textit{A.~thaliana};
the \cite{comeron2012many} map for \textit{D.~melanogaster},
and the \cite{campbell2016pedigree} map for \textit{C.~familiaris}.
Adding further maps to the library is straightforward.
The second key element of a species description
within \stdpopsim is a set of carefully curated population genetic model
descriptions from the literature, which allow simulation under
specific historical scenarios that have been fit to present-day patterns of
genetic variation (See the Methods for a description of the community
development and quality-control process for these models.)

The current demographic models in the \stdpopsim catalog are shown in \autoref{tab:catalog}.
These range from
simple, single population histories \cite[e.g.,][]{sheehan2016deep},
to complex models which include population splitting, migration, and archaic
admixture \cite[e.g.,][]{ragsdale2019models}.
In addition to models from Table \autoref{tab:catalog},
the PopSim Consortium has demographic models for \textit{Pongo abelii}, \textit{Canis familiaris}, and \textit{Escherichia coli} in development at time of writing.

\renewcommand{\arraystretch}{1.2}
\begin{table}[t]
% Make sure the table is centered even though it's too wide
\makebox[\textwidth][c]{
    \begin{footnotesize}
    \input{model_table.tex}
    \end{footnotesize}
}
\caption{\label{tab:catalog}
Initial set of demographic models in the catalog and summary of computing resources needed for simulation.
For each model, we report the CPU time, maximum memory usage and the
size of the output \tskit file, as simulated using the \texttt{msprime} simulation engine.
In each case, we simulate 100 samples
drawn from the first population, for the shortest chromosome of that species
and a constant chromosome-specific recombination rate.
The times reported are for a single run on an Intel i5-7600K CPU.
Computing resources required will vary widely depending on sample sizes, chromosome length,
recombination rates and other factors.
}
\end{table}

Currently, {\em Homo sapiens} has the largest number of population models in
\stdpopsim (see Table \ref{tab:catalog}).
These models include: a simplified version of the \cite{tennessen2012evolution}
model with only the African population specified (expansion from the ancestral
population and recent growth; Africa\_1T12); the three-population model of \cite{gutenkunst2009inferring},
which specifies the out-of-Africa bottleneck as well as the subsequent divergence of
the European and Asian populations (OutOfAfrica\_3G09); the \cite{tennessen2012evolution} two-population variant of the
\citeauthor{gutenkunst2009inferring} model, which does not include Asian populations but more explicitly models
recent rapid human population growth in Europe (OutOfAfrica\_2T12); the \cite{browning2018ancestry} admixture model
for American populations, which specifies ancestral African, European, and Asian population
components (AmericanAdmixture\_4B11); a three-population out-of-Africa model from \cite{ragsdale2019models},
which includes archaic admixture (OutOfAfricaArchaicAdmixture\_5R19);
a complex model of ancient Eurasian admixture from \cite{kamm2019efficiently} (AncientEurasia\_9K19);
and a synthetic model of oscillating population size from \cite{schiffels2014inferring} (Zigzag\_1S14).


For \textit{D.~melanogaster}, we have implemented the three-epoch model estimated by \cite{sheehan2016deep} from
an African sample (African3Epoch\_1S16), as well as the out-of-Africa divergence
and associated bottleneck model of \cite{li2006inferring}, which jointly models African
and European populations (OutOfAfrica\_2L06). For \textit{A.~thaliana}, we implemented the
model in \cite{durvasula2017african} inferred using \MSMC. This model includes
a continuous change in population size over time, rather than pre-specified epochs of different
population sizes (SouthMiddleAtlas\_1D17). We have also implemented a two-epoch and a three-epoch model estimated from African
samples of \textit{A.~thaliana} in \cite{huber2018gene} (African2Epoch\_1H18 and African3Epoch\_1H18).
%For \emph{P. pygmaeus} we have implemented the two
%species divergence model of \cite{locke2011comparative}, which models isolation with migration between \emph{P. pygmaeus}
%and \emph{P. abelii}.
%For \emph{E. coli} we have a simple, large constant population size model implemented.

In addition to organism-specific models, \stdpopsim also includes a generic piecewise constant size model and
isolation with migration (IM) model which can be used with any genome and genetic map.
Together these models
contain many features believed to affect observed patterns of polymorphism (e.g., bottlenecks, population growth,
admixture) and therefore provide useful benchmarks for method development.


%DRS: do we want to include these generic models in Table 1?

To guarantee reproducibility,
we have standardized naming
conventions for species, genetic maps, and demographic models that will enable long-term stability
of unique identifiers used throughout \stdpopsim,
as described in our documentation
(\url{https://stdpopsim.readthedocs.io/en/latest/development.html#naming-conventions}).

\subsection*{Simulation engines}

Currently, \stdpopsim uses the  \texttt{msprime} coalescent simulator \citep{kelleher2016efficient}
as the default simulation engine.
Coalescent simulations, while highly efficient, are limited in their ability to model
continuous geography or
complex selection scenarios, such as recurrent sweeps and background selection.
For these reasons,
we have also implemented the forward-time simulator, \texttt{SLiM} \citep{haller2019tree,haller2019slim},
as an alternative backend engine to \texttt{stdpopsim}, allowing for the simulation of processes that cannot be modeled under
the coalescent. However, as forward-time simulators explicitly model all individuals in a population, simulating large
population sizes can be highly demanding of computational resources. One common practice used to address this challenge is to
simulate a \emph{smaller} population, but to rescale resulting times,
mutation rates, recombination rates, and selection coefficients
so that the intensity of mutation, recombination, and allele frequency change due to selection
per unit time remains the same
%% PLR: the following is not worded quite right, or at least confusingly, because if you scale time
%% then this scales rates also; I've opted for a less precise but more descriptive formulation?
% scale down the population size used in the simulations (\textit{N}),
% scale down times specified in the demographic model by the same factor, and scale up the mutation and recombination
% rates and selection coefficients such that the scaled parameters, \textit{N$\mu$}, \textit{Nr}, and \textit{Ns}, remain unchanged
\citep[see the \texttt{SLiM} manual and][]{uricchio2014robust}.
Our implementation of the \texttt{SLiM} backend allows easy use of this \emph{rescaling}
through a single ``scaling factor'' argument.
Such down-scaled simulations are not completely equivalent to simulating
all individuals in the population, and may lead to subtle differences,
especially in the presence of selection.
However, since many sequence-based measures of population diversity remain nearly unchanged when rescaling in this fashion,
this practice is effective for many purposes and widely employed.

We validated our implementation of the \texttt{SLiM} engine by comparing
estimates of several population genetic summary statistics for neutral simulations generated by both \texttt{SLiM}
and \texttt{msprime}.
Examples of this validation for the AncientEurasia\_9K19 model
\citep{kamm2019efficiently}
are shown in Figures \ref{fig:slim_val_map} and \ref{fig:slim_val_nomap}.
For this model,
down-scaling factors of up to $50$ produce patterns of both diversity and linkage disequilibrium that are
very similar to those observed under the coalescent (i.e., \texttt{msprime}), when both are simulated using the HapMapII\_GRCh37 recombination map
(Figure \ref{fig:slim_val_map}),
although the distribution of Tajima's $D$ differs slightly when scaling down by a factor of $50$.
%likely due to the subtle differences produced by down-scaling.
Interestingly, the effects of down-scaling are more prominent and extend to other sequence-based statistics
when data sets are simulated under a uniform recombination rate (Figure \ref{fig:slim_val_nomap}).
Importantly, both comparisons validate the equivalence of \texttt{SLiM} and \texttt{msprime} when no down-scaling is applied.
The results are also optimistic about the rescaling strategy to reduce computational burden,
but the possible effects are not well-understood,
so results relying on rescaled simulations should be carefully validated.

\subsection*{Documentation and reproducibility}

The \stdpopsim command-line interface, by default, outputs citation information
for the models, genetic maps, and simulation engines used in any particular run.
We hope that this feature will encourage users to appropriately acknowledge the
resources used in published work, and encourage authors
publishing demographic models to contribute to our ongoing community-driven development process.
Together with the \stdpopsim version number and the long-term stable identifiers
for population models and genetic maps,
this citation information will result in well-documented and reproducible
simulation workflows. The individual tree sequence files produced by
\stdpopsim also contain complete provenance information including the command
line arguments, operating system environment and versions of key libraries
used.

\subsection*{Use case: comparing methods of demographic inference}

As an example of the utility of \stdpopsim, we demonstrate how it can be easily used
to perform a fair comparison of popular demographic inference methods.
Although we present comparison of results from several
methods, our aim at this stage is not to provide an exhaustive
evaluation or ranking of these methods. Our hope is instead to demonstrate how \stdpopsim
will facilitate more detailed future explorations of the strengths and weaknesses of the numerous
inference methods that are available to the population genetics community
(see Discussion).

We start by comparing popular methods for estimating
population size histories of single populations and subsequently
show simple examples of multi-population inference.
To reproducibly evaluate and compare the performance of inference methods, we developed
workflows using \texttt{snakemake} \citep{koster2012snakemake},
available from \url{https://github.com/popsim-consortium/analysis},
that allow efficient computing in multicore or cluster environments.
Our workflow generates $R$ replicates of $C$ chromosomes,
% ACS: "sample" could be confusing because we are both sampling
% by simulation and sampling from a population.  Does "population samples" (below) help?
producing $n$ population samples in each of a total of $R \times C$ simulations for each demographic model.
After simulation,
the workflow prepares input files for each inference method
by grouping all $n \times R \times C$ simulated chromosomes
into a single file.
Each file is then converted into an input file appropriate for each inference method
(such that all inference methods run on the same simulation replicates).
Each of the inference programs are then run in parallel, and finally,
estimates of population size history from each program are plotted.

% ACS: do we need the "Homo sapiens" at top left in the figure?  It breaks the visual symmetry among YRI, CEU, and CHB, and
% I think it is unnecessary given the population labels and the description in the legend
% Also: should we include "N(t)" after "Population size" in the y-axis label, to drive home its meaning (used in the text and legend)?
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_OutOfAfricaArchaicAdmixture_5R19.pdf}
\caption{\textbf{Comparing estimates of $N(t)$ in humans}. Here we show estimates of population
size over time ($N(t)$) inferred using 4 different methods: \smcpp, \texttt{stairway plot}, and
\MSMC with $n=2$ and $n=8$ samples. Data were generated by simulating
replicate human genomes under the OutOfAfricaArchaicAdmixture\_5R19 model \citep{ragsdale2019models} and using the
HapMapII\_GRCh37 genetic map~\citep{international2007second}. From top to bottom we show estimates for each
of the three populations in the model (YRI, CEU, and CHB). In shades of blue we show the estimated
$N(t)$ trajectories for each of three replicates.
As a proxy for the ``truth'', in black we show inverse coalescence rates
% perhaps add: "(expressed in numbers of individuals)"    Seems useful to make the units clear
as calculated from the demographic model used for simulation (see text).
}
\label{fig:n_t_ragsdale}
\end{center}
\end{figure}


% ACS: there was indeed a lot of redundancy here.  DRS improved it a lot, but I've tried to do a bit more
\subsubsection*{ Single-population demographic models.}
For single-population demographic models, we compared
\MSMC~\citep{schiffels2014inferring}, \smcpp~\citep{terhorst2017robust}, and
\stairwayplot~\citep{liu2015exploring}
 on simulated genomes sampled from a single population,
under several of the demographic models described above.
However, these experiments raise the question of what to use as the ``true" population sizes in the case of multi-population models with migration.
In particular, a simple single-population model that is fit to data simulated
under a multi-population model,
is not expected to recover the actual simulated population sizes,
because of model misspecification.
Instead, we argue that the best one may expect in such a scenario is to infer a model that accurately reflects the
coalescence time distribution of the simulated model.
Under a multi-population model, the coalescence time distribution is influenced by migration between the target population and populations not analyzed in inference,
as well as by the ancestral effective population sizes.
The inverse coalescence rate is commonly interpreted as the effective population size,
since these are equal in a single-population model with random mating.
We thus analytically computed inverse coalescence
rates in \texttt{msprime} for each simulated model, and used them as benchmarks for the ``true'' effective population sizes.
See the Appendix for a precise definition and description of the inverse coalescence rate computation.

Figure \ref{fig:n_t_ragsdale} presents the results
from simulations under OutOfAfricaArchaicAdmixture\_5R19,
a model of human migration out of Africa that includes archaic admixture
\citep{ragsdale2019models}, along with an empirical genetic map. In each column of this figure
we show the inferred population size history (denoted $N(t)$)
from samples taken from each of the three extant populations in the model.
In each row we show comparisons among the methods (including two sample sizes for \MSMC).
Blue lines show estimates from each of three replicate whole genome simulations,
and black lines indicate the ``true" values depicted by the inverse coalescence rates (although in this specific model
the inverse coalescence rates are very close to the simulated population sizes;
Figure \ref{fig:census_vs_IRC}).
%DRS: trying to cut down on the redundancy between the coal rate stuff here and the previous paragraph
While there is variation in accuracy among methods, populations, and individual replicates,
the methods generally produce a good estimate
of the true effective population sizes of the simulations, with
inferred values mostly within a factor of two of the truth,
and most methods inferring a bottleneck at approximately the correct time.

Using \stdpopsim, we can readily compare performance on this benchmark to
that based on a different model of human history. In Figure \ref{fig:n_t_gutenkunst} we show estimates of
$N(t)$ from simulations using the same physical and genetic maps, but from the OutOfAfrica\_3G09
demographic model that does not include archaic admixture. Again we see that each
of the methods is capturing relevant parts of the population history, although the
accuracy varies across time. In comparing inferences between the
models it is interesting to note that $N(t)$ estimates for the CHB and CEU
simulated populations are generally better across methods than estimates from the YRI
simulated population.

We can also
see how well methods might do at recovering the population history of a constant-sized population,
with human genome architecture and genetic map.
We show results of such an
experiment in Figure \ref{fig:n_t_HomSap_constant}.
All methods recover population size within a factor of two of the simulated values, however
SMC-based methods
% ACS: suggest eliminating this vague conjecture.  Which regularization exactly is meant?
%perhaps due to their regularization,
tend to infer sinusoidal
patterns of population size even though no such change is present.


\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/DroMel_African3Epoch_1S16.pdf}
\caption{\textbf{Comparing estimates of $N(t)$ in \textit{Drosophila}}. Population
size over time ($N(t)$) estimated from an African population sample. Data were generated by simulating
replicate \textit{D.~melanogaster} genomes under the African3Epoch\_1S16 model \citep{sheehan2016deep}
with the genetic map of \cite{comeron2012many}. In shades of blue we show the estimated
$N(t)$ trajectories for each replicate.
As a proxy for the ``truth'', in black we show inverse coalescence rates
as calculated from the demographic model used for simulation (see text).
}
\label{fig:n_t_sheehan}
\end{center}
\end{figure}

As most method development for population genetics has been focused on human
data, it is of consequence to ask how such methods might perform in non-human
genomes. Figure \ref{fig:n_t_sheehan} shows parameter estimates from the African3Epoch\_1S16
model, originally estimated from an African sample of \textit{D.~melanogaster} \citep{sheehan2016deep},
and Figure \ref{fig:n_t_AraTha} shows estimates from simulations of \textit{A.~thaliana}
under the African2Epoch\_1H18 model originally inferred by \cite{huber2018gene}.
In both cases, as with humans, we use \stdpopsim to simulate replicate genomes using an empirically-derived genetic map,
and try to infer back parameters of the simulation model.
Accuracy is mixed among methods when doing inference on simulated data from these \textit{D.~melanogaster}
and \textit{A.~thaliana} models, and generally worse than what we
observe for simulations of the human genome.


\begin{figure}
\begin{center}
\includegraphics[width=0.7\linewidth]{display_items/homo_sapiens_two_pop_comp.pdf}
\caption{\textbf{Parameters estimated using a multi-population human model}.
Here we show estimates of $N(t)$ inferred using \dadi, \fastsimcoal, and \smcpp. \textbf{(A)} Data were generated by simulating
replicate human genomes under the OutOfAfrica\_3G09 model and using the HapMapII\_GRCh37 genetic map
inferred in \cite{international2007second}. \textbf{(B)} For \dadi and \fastsimcoal we show parameters inferred
by fitting the depicted IM model, which includes population sizes, migration rates, and a split
time between CEU and YRI samples. \textbf{(C)} Population size estimates for each population (rows)
from \dadi, \fastsimcoal, and \smcpp (columns).
In shades of blue we show $N(t)$ trajectories estimated from each simulation,
and in black simulated population sizes for the respective population.
The population split time, $T_{DIV}$, is shown at
the bottom (simulated value in black and inferred values in blue),
with a common $x$-axis to the population size panels.}
\label{fig:IM_popn_human}
\end{center}
\end{figure}

\subsubsection*{Multi-population demographic models.}
As \stdpopsim implements multi-population demographic models, we also
explored parameter estimation of population divergence parameters. In particular,
we simulated data under multi-population models for humans and \textit{D.~melanogaster}
and then inferred  parameters using \dadi, \fastsimcoal, and \smcpp.
For simplicity, we conducted inference in \dadi and \fastsimcoal by fitting an isolation with migration (IM) model
with constant population sizes and bi-directional migration \citep{hey2004im}.
Our motivation for fitting this simple
IM model was to mimic the typical approach of two population inference on empirical
data, where the user is not aware of the `true' underlying demography and the inference
model is often misspecified.
For human models with more than two populations \citep[e.g.,][]{gutenkunst2009inferring}
this limitation means that users are inferring parameters for a model that does
not match the model from which the data were generated (Figures
\ref{fig:IM_popn_human}A and B). However, since the model used for inference also
allows gene flow between populations, we directly compare estimated effective population sizes
to the values used in simulations (black line in Figure \ref{fig:IM_popn_human}C)
and not the inverse coalescence rates.

In Figure \ref{fig:IM_popn_human}C we show estimates of population sizes and divergence
time, for each of the inference methods, using samples drawn from African and European populations
simulated under the OutOfAfrica\_3G09 model. Our results highlight many
of the strengths and weaknesses of the different methods.
For instance, the SFS-based approaches with simple IM models do not capture
recent exponential growth in the CEU population, but do consistently recover the
simulated YRI population size history. Moreover, these approaches allow
migration rates to be estimated (Figure \ref{fig:homsap_mig_rates}), and lead to more accurate inferences
of divergence times. However, these migration rate estimates are somewhat biased.
In contrast, \smcpp is much better at capturing the recent exponential
growth in the CEU population, though it consistently underestimates divergence times
because it assumes no migration between populations (Figure \ref{fig:IM_popn_human}C).

Again, we can extend this analysis to other taxa and examine the performance of these methods
for a two-population model of \textit{D.~melanogaster}. Figure
\ref{fig:two_popn_fly} shows inference results using data simulated under
the OutOfAfrica\_2L06 model. This model includes an ancestral population in
Africa from which a European population splits off following a bottleneck, with no
post-divergence gene flow between the African and European population (Figure \ref{fig:two_popn_fly}A).
Here again, we find that \dadi and \fastsimcoal infer more consistent histories,
but they do not detect the brief bottleneck in Europe, due to the inference model
not allowing for population size changes after the population split. In addition, \dadi
and \fastsimcoal both do reasonably well at correctly inferring the absence of
migration (Figure \ref{fig:dromel_mig_rates}). In contrast, the inferred
demographic parameters from \smcpp are more noisy, though in some cases better
capture the short bottleneck in the European population.

Although these results do not represent an exhaustive benchmarking,
we have highlighted some of the strengths and weaknesses of these methods.
Future work should build on these results and undertake more in-depth comparisons
under a wider range of simulated demographic models.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Discussion}

Here we have described the first major product from the PopSim Consortium:
the \stdpopsim library. We have founded the Consortium with a number of specific goals in mind:
standardization of simulation within the population genetics community,
increased reproducibility and ease of use of complex simulations,
community-based development and decision making guiding best practices in population genetics,
and benchmarking of inference methods.

The \stdpopsim library allows for rigorous
standardization of complex population genetic simulations. Population genetics, as a field,
has yet to coalesce around a set of standards for the crucial task of method
evaluation, which in our discipline hinges on simulation. In contrast, other fields such as
structural biology \citep{moult1995large} and machine learning \citep{russakovsky2015imagenet} have a long track record
of standardized method testing. We hope that our efforts represent the beginning of what
will prove to be an equally longstanding and valuable tradition in population genetics.

Besides being a resource for developers of computational methods,
we aim for \stdpopsim to be a resource for empirical researchers using genomic data.
For instance, \stdpopsim could be used in power analyses to determine adequate sample sizes,
or in sanity checks to see if observed data
(e.g., levels of divergence or the allele frequency spectrum)
are roughly consistent with the hypothesized scenario.
Currently, many studies would benefit from such simulation-based checks.
However,  there are major barriers to implementation,
since individual research groups must reimplement complex, previously published demographic models,
a task made especially daunting by additional layers of realism (e.g., recombination maps).

\paragraph{Benchmarking population size inference.}
We have illustrated in this paper how \stdpopsim can be used for direct
comparisons of inferential methods on a common set of simulations. Our
benchmarking comparisons have been limited, but nevertheless
reveal some informative features.
For example, at the task of estimating population size histories for simulated human
populations, we find that the sequence-based methods (\MSMC and \smcpp)
perform somewhat better overall---at least for moderate times in the past---than
the site frequency spectrum-based method (\stairwayplot),
which tends to over-estimate the sizes of oscillations
(Figures~\ref{fig:n_t_ragsdale} and~\ref{fig:n_t_gutenkunst}).
In contrast, \stairwayplot outperforms the sequence-based methods
on simulations of \textit{D.~melanogaster} or \textit{A.~thaliana} populations,
in which linkage disequilibrium is reduced (Figures~\ref{fig:n_t_sheehan} \& \ref{fig:n_t_AraTha}).
In simulations of two human populations
(Figure~\ref{fig:IM_popn_human}), \dadi and \fastsimcoal do reasonably well at
reconstructing the simulated YRI history and estimating divergence times, but struggle with the more complex simulated CEU
history, in large part because the methods assume constant population sizes.
On the other hand, \smcpp does not have the same
restrictions on its inferred history, and as a result does much better
with the CEU history but tends to underestimate divergence times
due to the assumption of no migration.
The results for the two-population \textit{D.~melanogaster} model (Figure~\ref{fig:two_popn_fly}) are generally similar.
In these comparisons, \fastsimcoal and \dadi perform almost identically, which is expected because they fit the same models to the same summaries of the data,
differing only in how they calculate model expectations and optimize parameters.

All methods for inferring demographic history have strengths and weaknesses \citep[as recently reviewed by][]{beichman2018review}.
We compared inferences from simulated whole genome data, but many factors affect choice of methodology.
Markovian coalescent methods (\MSMC and \smcpp) require long contiguous stretches of sequence data.
In contrast, frequency spectrum methods (\stairwayplot, \dadi, and \fastsimcoal) can use reduced-representation sequencing data, such as RADseq \citep{andrews2016radseq}.
\dadi and \fastsimcoal require a pre-specified parametric model, unlike \MSMC, \smcpp, and \stairwayplot.
Using a parametric approach yields less noisy results, but a model that is too simple may not capture important demographic events (Figures~\ref{fig:IM_popn_human} and~\ref{fig:two_popn_fly}),
and other forms of model misspeficification may also produce undesireable behavior.
From a software engineering perspective, methods also differ in their ease of installation and use.
We hope our workflows will assist in the application of all the methods we have considered.

Altogether, these preliminary experiments highlight
the utility of \stdpopsim for comparing a variety of inference methods on
the same footing, under a variety of different demographic models.
In addition, the ability of \stdpopsim to generate data with and without significant features, such
as a genetic map or population-size changes (e.g., Figure \ref{fig:n_t_HomSap_constant}), allows
investigation of the failure modes of popular methods.
Moreover the comparison of methods across the various genome organizations, genetic maps,
and demographic histories of different organisms, provides valuable information
about how methods might perform on non-human systems.
Finally, comparison of results across methods or simulation runs
provides an estimate of inference uncertainty, analogous to parametric bootstrapping,
especially when different methods are vulnerable to model misspecification in different ways.

\paragraph{Next steps.}
\Stdpopsim is intended to be a fully open, community-developed project.
Our implementations of genome representations and genetic maps for the some of
the most common study systems in computational genetics---humans, \textit{Drosophila},
and \textit{Arabidopsis} (among others)---are only intended to be a starting point for
future development.
Researchers are invited to contribute to the resource by adding their
organisms and models of choice. The \stdpopsim resource is
accompanied by clearly documented standard operating procedures that are
intended to minimize barriers to entry for new developers.  In this way, we
expect the resource to expand and adapt to meet the evolving needs of the
population genomics community.

One of our goals is to engage research communities studying other taxa,
so as to expand the resource to many more species.
Although we have included demographic models and recombination maps,
there are many biological processes that we do not model.
Some of the additions that we are enthusiastic to add are:
selection (including distributions of fitness effects, maps of functional elements, and both single and recurrent hitchhiking events),
gene conversion, mutation models (rate heterogeneity),
more realistic demography (overlapping generations, separate sexes, mortality/fecundity schedules),
geographic population structure,
and downstream aspects of data quality (genotyping and mapping error).
Moreover, an in-depth investigation into the effects of population-size rescaling under many of the
above scenarios is warranted, given our preliminary findings using neutral simulations
(Figures \ref{fig:slim_val_map} and \ref{fig:slim_val_nomap}).
These plans would not be possible without recent important work on SLiM \citep{haller2019slim},
as well as current work on \texttt{msprime}.
Some other important processes are more challenging to model with current simulation software,
such as
structural variation,
changing recombination maps over time,
transposable elements,
and context-dependent mutation.

% note: this is addressing reviewer comments about model uncertainty
In particular, although the included demographic histories are some of the most widely used models for our current set of species,
we anticipate the set of available models to expand
as new methods and new modeling frameworks are developed.
For instance, the current models all describe a small set of discrete, randomly mating populations,
which are likely good approximations for deep-time population history,
but may be less useful for methods describing dynamics of contemporary populations.
\Stdpopsim's framework is sufficiently general that more realistic population models
will be easily incorporated, once they are published.
%DRS: trying to make this part a little less subjective. We were kind of painting the picture that
%we have all the best/useful models and everything currently published but not included is not useful,
%and I don't think this is justified.
Our current set may also omit other widely used models that can be readily incorporated into \stdpopsim.
Additional complications such as changing
genome builds as assemblies improve must also be incorporated.
As the collection of models and species grows, we hope to
incorporate crowd-sourced community input
to highlight the most appropriate models and genome annotations.
%DRS: not in love with the word "annotations" here as some might think we are talking gene annotations (which we will be eventually but not yet), but "details" is too vague.


% \paragraph{The problem of comparison} % Moved from the intro
% Benchmarking the performance of inference methods in population genetics is not straightforward.
% Different methods often fit different types of model
% or use different summaries of the genetic variation data, even when
% attempting to address the same biological question. For example, there are
% numerous methods to infer population size changes.
% The \MSMC method \citep{schiffels2014inferring} allows for the
% population to continuously change in size over time. Other methods, such as
% \dadi \citep{gutenkunst2009inferring}, allow for a smaller number of size changes to occur at certain
% points throughout the population's history. The types of data that the
% methods use also differs. While \MSMC uses one to four whole genome sequences and leverages the
% spatial patterns of genetic variation along the sequence, \dadi  instead uses the
% site frequency spectrum (SFS), which summarizes the frequencies of SNPs in the
% sample, ignoring any information about the correlation structure (linkage
% disequilibrium) among SNPs. Thus from the start these methods may not be
% expected to perform equally well in all scenarios,
% and indeed, it is not always clear how to compare to the truth --
% for instance, how well can a method of piecewise constant population sizes
% describe a period of exponential growth?
% Or, how should discrete population models be mapped onto real-world (continuous) geography?
% The answer depends on determining what is most important for the question at hand.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Methods}
%\subsection*{Generating the PopSim resource}
% RNG: No need to empty subsection.

\subsection*{Model quality control}
As a consortium we have agreed to a standardized procedure for model inclusion
into \stdpopsim that allows for rigorous quality control. Imagine Developer A
wants to introduce a new model into \stdpopsim. Developer A implements the
demographic model for the relevant organism along with clear documentation
of the model parameters and populations. This model is submitted as a ``pull request'',
where it is evaluated by a reviewer and then included as `preliminary',
but is not linked to the online documentation nor the command line interface.
Developer A submits a quality control (QC) issue, after which a second developer,
Developer B (perhaps found by requesting review from the broader Consortium),
then independently reimplements the model from the relevant
primary sources and adds an automatic unit test for equality between the
QC implementation and the preliminary production model.
If the two implementations are equivalent, the original model is included in \stdpopsim.
If not, we move to an arbitration process whereby A and B first try
to work out the details of what went wrong. If that fails, the original
authors of the published model must be contacted
to resolve ambiguities.
% This process can lead to cycles of revision of the implementation from Developers A and B but allows the project to assure quality.
% Indeed we have already had a number of interesting cases of arbitration to this point.
Further details of our QC process can be found in our developer documentation
(\url{https://stdpopsim.readthedocs.io/en/latest/development.html#}).

The possibility for error and the importance of careful qualty control
was inadvertantly illustrated during our own development process:
during the final revisions of this paper, we noticed that the
OutOfAfrica\_3G09 model \citep{gutenkunst2009inferring}
had not in fact gone through the QC process.
The subsequent QC process revealed that in fact our implementation
slightly wrong---migration rates had not been set to zero to the European population in the longest-ago time period
when there should have only been a single population.
This error was propagated from the \texttt{msprime} documentation,
where the model was presented as an illustrative example.
A number of studies have been published using copies of
this example code.

\subsection*{Workflow for analysis of simulated data}
To demonstrate the utility of \stdpopsim we created \texttt{Snakemake}
workflows \citep{koster2012snakemake} that perform demographic inference on
tree sequence output from our package using a few common software packages (see Figure \ref{fig:n_t_dag} for an example workflow).
Our choice of \texttt{Snakemake} allows complete reproducibility of the
analyses shown, and all code is available from \url{https://github.com/popsim-consortium/analysis}.

We performed two types of demographic inference.
Our first task was to infer effective population size over time (denoted $N(t)$).
This was done using three software packages: \stairwayplot,
which uses site frequency spectrum information only \citep{liu2015exploring};
\MSMC~\citep{schiffels2014inferring},
which is based on the sequentially Markovian coalescent (SMC),
run with two different sample sizes ($n = 2 , 8$);
and \smcpp~\citep{terhorst2017robust},
which combines information from the site frequency spectrum with
recombination information as in SMC-based methods. No attempt
was made at trying to optimize the analysis from
any particular software package,
as our goal was not to benchmark performance of methods but
instead show how such benchmarking could be easily done using
the \stdpopsim resource. In this spirit we ran each software package as near
to default parameters as possible. For \stairwayplot we
set the parameters \texttt{numRuns=1} and \texttt{dimFactor=5000}.
For \smcpp we used the ``estimate" run mode to infer $N(t)$ with all other parameters set
to their default values. For \MSMC we used the \texttt{--fixedRecombination}
option and used the default number of iterations.

For the single-population task we ran human (HomSap) simulations
using a variety of models (see Table 1): OutOfAfricaArchaicAdmixture\_5R19, OutOfAfrica\_3G09,
and a constant-sized generic model.
% ILAN REMOVED THIS APR 29, 2020 - NOT USED ANYMORE
%, and a two-epoch generic model where the population size instantaneously decreased from $N=10^4$ to $N=10^3$
% five hundred generations before the present.
Each simulation
used the HapmapII\_GRCh37 genetic map. For \textit{D.~melanogaster}
we estimated $N(t)$ from an African sample simulated under the DroMel,
African3Epoch\_1S16 model using the Comeron2012\_dm6 map.
Finally, we ran simulations of \textit{A.~thaliana} genomes using the
AraTha African2Epoch\_1H18 model under the Salome2012\_TAIR7 map.
For each model, three replicate whole genomes were
simulated and the population size estimated from those data. In all cases we
set the sample size of the focal population to $N=50$ chromosomes.

Following simulation, low-recombination portions of chromosomes were masked
from the analysis in a manner that reflects the ``accessible" subset of sites
used in empirical population genomic studies \citep[e.g.,][]{danecek20111000,langley2012genomic}.
Specifically we masked all regions of \SI{1}{cM} or greater in the lowest 5th percentile of the empirical
distribution of recombination, regions which are nearly uniformly absent for empirical analysis.
This approach to masking was chosen to prevent marginal trees with low or no recombination
from biasing the comparisons of demographic inference methods. It should be noted that masking is not implemented
within \texttt{stdpopsim} proper; tree sequences generated by \texttt{stdpopsim} are always raw and unmasked.
This allows users the flexibility to implement masking approaches that are specific to their needs
for downstream analysis.

Our second task was to explore inference with two-population models
using some of the multi-population demographic models implemented in \stdpopsim.
For HomSap we used the OutOfAfrica\_3G09 model with the HapmapII\_GRCh37 genetic map,
and for DroMel we used the OutOfAfrica\_2L06 model with the Comeron2012\_dm6 map.
The HomSap model is a three population model (Africa, Europe, and Asia) including post-divergence
migration and exponential growth (Figure \ref{fig:IM_popn_human}C), whereas the
DroMel model is a two population model (Africa and Europe) with no post-divergence
migration and constant population sizes (Figure \ref{fig:two_popn_fly}).

To conduct inference on these models, we applied three commonly used methods:
\dadi~\citep{gutenkunst2009inferring}, \fastsimcoal~\citep{excoffier2013robust},
and \smcpp~\citep{terhorst2017robust}. As above, these methods were used
generally with default settings and we did not attempt to optimize their performance or fit
parameter-rich demographic models.

For both \dadi and \fastsimcoal, we fit a two population
isolation-with-migration (IM) model with constant population sizes.
This IM model contains six parameters:
the ancestral population size, the sizes of each population after the split,
the divergence time, and two migration rate parameters.
Importantly, this meant that for both species, the
fitted model did not match the simulated model (Figures \ref{fig:IM_popn_human} and \ref{fig:two_popn_fly}).
In the HomSap case, we therefore performed inference solely on the Africa
and Europe populations, meaning that the Asia population functioned as a ``ghost''
population that was ignored by our inference.
To validate our inference approach, we also conducted
inference on a generic IM model that was identical to the model used for inference (Figure \ref{fig:generic_IM}).

From HomSap simulations we took 20 whole genome samples each from the Europe and Africa populations
from each replicate.
Runtimes of DroMel simulations were prohibitively slow when simulating whole genomes
with the Comeron2012\_dm6 map due to large effective population sizes leading to
high effective recombination rates. For this reason, we present only data
from 50 samples of a \SI{3}{\mega\byte} region of chromosome 2R from simulations under OutOfAfrica\_2L06.
For the generic IM simulations, we used the HomSap genome along with the
HapmapII\_GRCh37 genetic map and sampled 20 individuals from each population.

Following simulation, we output tree sequences and masked low-recombination
regions using the same approach described for the single population workflow above. We
converted tree sequences into a two-dimensional site frequency spectrum for all
chromosomes in the appropriate format for \dadi and \fastsimcoal.
For each simulation
replicate, we performed 10 runs of \dadi and \fastsimcoal, checking to ensure that
each method reached convergence.

Detailed settings for \dadi and \fastsimcoal can be found in the Snakefile
on our git repository (\url{https://github.com/popsim-consortium/analysis}).
Estimates from the highest log-likelihood (out of 10 runs) for each simulation replicate
are shown in Figures \ref{fig:IM_popn_human}C and \ref{fig:two_popn_fly}C.

For \smcpp, we converted the tree sequences into VCF format and performed inference
with default settings. Importantly, \smcpp assumes no migration post-divergence,
deviating from the simulated model.  However, because \smcpp allows
for continuous population size changes, it is better equipped to capture many of the
more complex aspects of the simulated demographic models (e.g., exponential growth).

To visualize our results, we plotted the inferred population size trajectories
for each simulation replicate alongside the simulated population sizes
(Figures \ref{fig:IM_popn_human}C and \ref{fig:two_popn_fly}C).
Here, unlike the single-population workflow,
we compare our inferred population sizes only to the simulated population
sizes and not the inverse coalescence rates.

\section*{Resource availability}
The version 0.1 release of \stdpopsim is available for download on Github:
\url{https://github.com/popsim-consortium/stdpopsim/releases}.
Documentation for the project can be found here:
\url{https://stdpopsim.readthedocs.io/en/latest/}.




% \subsection*{Extra text}
% [This section describes the stdpopsim Python library. What it’s for, how it’s used and what it can do]
% [First para: running accurate simulations is hard]
% Running realistic population genetic simulations is a complex task today. It is
% straightforward to run simple simulations of population models such as the
% coalescent or Wright-Fisher using (e.g.) msprime or SLiM, but the process of
% instantiating these models to accurately reflect the properties of natural
% populations that are important for a particular study is fraught with
% difficulties. There are typically three important inputs that must be decided:
% the demographic model, recombination and mutation rates. Demographic models for
% a particular species are typically chosen by examining the literature for
% published model specifications, which can consist of dozens of parameters. These
% parameters must be transcribed from the original publication and translated into
% the input format required for the simulator in question. This is a difficult and
% error prone task, and mistakes are common. Recombination and mutation rates are
% simpler to describe and often a single genome-wide estimate is appropriate.
% However, such estimates are updated frequently and there is wide variation in
% the values used for the simulations of the same species. When more fine-grained
% simulations of the local chromosome structure is required, empirical
% recombination and mutation rate maps are used. However, there is no standard
% format for such maps, and finding the maps can be challenging.

\section*{Acknowledgments}
We thank the Probabilistic Modeling in Genomics conference organizers for making this collaboration possible,
and the Simons Center for Quantitative Biology at Cold Spring Harbor Laboratory
for sponsoring the first workshop.
Early on in the project we were encouraged by many people including Patrick Phillips, Richard Durbin,
Dmitri Petrov, and Sohini Ramachandran.
CCK and KEL were funded under NIH Award R35GM119856.
JRA and ADK were funded under NIH Award R01GM117241.
TJS and RNG were funded under NIH Award R01GM127348.
ALG and DRS were funded under NIH award R00HG008696.
ND and AS were supported in part by NIH Awards R01HG010346 and R35GM127070.
FR and GG were supported by a Villum Young Investigator award (project no.~00025300).
DODV is funded by a UC MEXUS-CONACYT Collaborative Grant and a DGAPA-PAPIIT grant (PAPIIT-IA200620).
JK is supported by the Robertson Foundation.


\bibliographystyle{plainnat}
\bibliography{popSim.bib}

\pagebreak
\beginsupplement
\section*{Supplemental Figures}
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_AE9K19_chr22_map_AI.pdf}
\caption{\textbf{Validating the \texttt{SLiM} engine backend under a genetic map}.
Here we validate our integration of the \texttt{SLiM} \citep{haller2019tree,haller2019slim} engine backend.
We show quantile-quantile plots between \texttt{SLiM} and \texttt{msprime} engines for three
population genetic summary statistics: \textit{r}\textsuperscript{2}, Tajima's $\pi$, and Tajima's D. Additionally, we
show runtimes for generating each simulation replicate. Data were generated by simulating 100 replicates
of human chromosome 22 under the AncientEurasia\_9K19 model \citep{kamm2019efficiently} and using the
HapMapII\_GRCh37 genetic map \citep{international2007second}. From top to bottom we show results
using three scaling factors for the population sizes: Q=1, Q=10, and Q=50.
}
\label{fig:slim_val_map}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_AE9K19_chr22_nomap_AI.pdf}
\caption{\textbf{Validating the \texttt{SLiM} engine backend under uniform recombination}.
Here we validate our integration of the \texttt{SLiM} \citep{haller2019tree,haller2019slim} engine backend.
We show quantile-quantile plots between \texttt{SLiM} and \texttt{msprime} engines for three
population genetic summary statistics: \textit{r}\textsuperscript{2}, Tajima's $\pi$, and Tajima's D. Additionally, we
show runtimes for generating each simulation replicate. Data were generated by simulating 100 replicates
of human chromosome 22 under the AncientEurasia\_9K19 model \citep{kamm2019efficiently} and using a uniform
rate of recombination across the chromosome. From top to bottom we show results
using three scaling factors for the population sizes: Q=1, Q=10, and Q=50.
}
\label{fig:slim_val_nomap}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_OutOfAfricaArchaicAdmixture_5R19_census_vs_IRC.pdf}
\caption{\textbf{Comparing simulated population sizes and inverse coalescence rates in humans}.
Data are shown from human genomes under the OutOfAfricaArchaicAdmixture\_5R19 model \citep{ragsdale2019models}
and using the HapMapII\_GRCh37 genetic map \citep{international2007second}.
From left to right we show sizes for each of the three populations in the model: YRI, CEU, and CHB.
We plot the simulated sizes for each population in black, and in red we plot
inverse coalescence rates as calculated from the demographic model used for simulation (see text).
In this specific model, these two measures are near identical, but in other models with higher migration rates
we expect to see a larger departure between the two.
}
\label{fig:census_vs_IRC}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_OutOfAfrica_3G09.pdf}
\caption{\textbf{Comparing estimates of $N(t)$ in humans}. Estimates of population
size over time ($N(t)$) inferred using 4 different methods, \smcpp, \stairwayplot, and
\MSMC with $n=2$ and $n=8$. Data were generated by simulating
replicate human genomes under the OutOfAfrica\_3G09 model \citep{gutenkunst2009inferring}
and using the HapMapII\_GRCh37 genetic map \citep{international2007second}.
From top to bottom we show estimates for each of the three populations in the model:
YRI, CEU, and CHB. In shades of blue we show the estimated
$N(t)$ trajectories for each replicate.
As a proxy for the ``truth'', in black we show inverse coalescence rates
as calculated from the demographic model used for simulation (see text).
}
\label{fig:n_t_gutenkunst}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/HomSap_constant_size.pdf}
\caption{\textbf{Comparing estimates of $N(t)$ in humans}. Here we show estimates of population
size over time ($N(t)$) inferred using 4 different methods, \smcpp, and \texttt{stairway plot}, and
\MSMC with $n=2$ and $n=8$. Data were generated by simulating
replicate human genomes under a constant sized population model with $N=10^4$
and using the HapMapII\_GRCh37 genetic map \citep{international2007second}.
As a proxy for the ``truth'', in black we show inverse coalescence rates
as calculated from the demographic model used for simulation (see text).
}
\label{fig:n_t_HomSap_constant}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/AraTha_African2Epoch_1H18.pdf}
\caption{\textbf{Comparing estimates of $N(t)$ in \textit{A.~thaliana}}. Here we show estimates of population
size over time ($N(t)$) inferred using 4 different methods, \smcpp, and \texttt{stairway plot}, and
\MSMC with $n=2$ and $n=8$. Data were generated by simulating
replicate \textit{A.~thaliana} genomes under the African2Epoch\_1H18 model \citep{durvasula2017african}
and using the SalomeAveraged\_TAIR7 genetic map \citep{salome2011recombination}.
As a proxy for the ``truth'', in black we show inverse coalescence rates
as calculated from the demographic model used for simulation (see text).
}
\label{fig:n_t_AraTha}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/homsap_migration_rates.pdf}
\caption{\textbf{Migration rate estimates for the human Gutenkunst model.}
Here we show inferred migration rates from \dadi and \fastsimcoal.
Data were generated by simulating
replicate human genomes under the \cite{gutenkunst2009inferring} model and using the genetic map
inferred in \cite{international2007second}.
Directional migration from Europe to Africa is represented as $MIG\_AF\_EU$ and
migration from Africa to Europe is represented as $MIG\_EU\_AF$.
Note that the $x$-axis coordinates are arbitrary.}
\label{fig:homsap_mig_rates}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/d_mel_two_popn_comp.pdf}
\caption{\textbf{Parameters estimated using a two-population \emph{Drosophila} model}.
Here we show estimates of $N(t)$ inferred using \dadi, \fastsimcoal, and \smcpp.
Data were generated by simulating
replicate \emph{Drosophila} genomes under the \cite{li2006inferring} model and using the genetic map
inferred in \cite{comeron2012many}. See legend of Figure \ref{fig:IM_popn_human} for details.
In shades of blue we show the estimated
$N(t)$ trajectories for each replicate. In black we show the simulated population sizes.}
\label{fig:two_popn_fly}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/dromel_migration_rates.pdf}
\caption{\textbf{Migration rate parameters estimated under a two-population \emph{Drosophila} model}.
Here we show inferred migration rates from \dadi and \fastsimcoal.
Data were generated by simulating replicate \emph{Drosophila} genomes under the \cite{li2006inferring} model and using the genetic map
inferred in \cite{comeron2012many}.
Directional migration from Europe to Africa is represented as $MIG\_AF\_EU$ and
migration from Africa to Europe is represented as $MIG\_EU\_AF$.
Note that the $x$-axis coordinates are arbitrary.}
\label{fig:dromel_mig_rates}
\end{center}
\end{figure}



\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/n_t_dag.pdf}
\caption{\textbf{Workflow for our N(t) inference methods comparison.}
Here we show single replicate for two chromosomes, chr22 and chrX, simulated under the HomSap
OutOfAfrica\_3G09 demographic model, with a HapmapII\_GRCh37 genetic map.
Note that the data used as input by all inference methods \smcpp, \MSMC, and \stairwayplot,
come from the same set of simulations.}
\label{fig:n_t_dag}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{display_items/generic_IM.pdf}
\caption{\textbf{Parameters estimated from a generic IM model} Here we show estimates of $N(t)$
inferred using \dadi, \texttt{fastsimcoal2}, and \smcpp. Data were generated by simulating
under a generic IM model with a human genome and \cite{international2007second} genetic map.
In shades of blue we show the estimated
$N(t)$ trajectories for each replicate. In black we show the simulated population sizes.}
\label{fig:generic_IM}
\end{center}
\end{figure}



\stopsupplement

\appendix

\section*{Appendix: Calculating coalescence rates}

In population genetics, the ``effective population size''
of a population model with constant (census) size
is often defined to be the number of diploids in a Wright-Fisher population
that would have the same coalescence rate (or, equivalently, genetic drift)
as the population in question \citep[reviewed in][]{crow1988inbreeding}.
One reason the concept is useful
is because theory predicts that genetic data from distinct populations
with the same effective population size will look similar in many ways:
for instance, their mean coalescence times will be the same.
Conversely, this implies that effective population size
should be easier to infer from genomic data than aspects of population demography
that do not affect effective population size.
An analogous observation holds for populations of changing size,
if we define the ``coalescence rate" of a given demographic model
at a particular point back in time
to be the rate of coalescence of remaining lineages
and define the ``coalescence effective size" at that time,
denoted $N_e(t)$, so that the coalescence rate at time $t$ in the past
is $1/(2N_e(t))$.
With these definitions, any two models with the same effective population size trajectory ($N_e(t)$)
will have the same \emph{distribution} of coalescence times.
For this reason,
we might guess that if we apply an inference method
that assumes a Wright-Fisher population with changing size through time
to a different population model,
the inferred demographic history will match the ``effective population size history''
defined in this way.
These observations and the following calculations are standard in coalescent theory
\citep[see e.g.,][]{wakeley2005coalescent},
but they are provided here for completeness.

We compute the coalescence rate of a collection of samples in a given demographic model
at a particular point back in time
as the expected number of coalescences happening at that time
per unit of time and per pair of as-yet-uncoalesced lineages.
More concretely,
let $p(t)$ denote the probability that the lineages of a randomly chosen pair of samples
have not yet coalesced $t$ units of time ago,
let $p(z, t)$ denote the probability that those lineages have not yet coalesced
and are furthermore both in location $z$,
and let $N_e(z,t)$ be the (effective) diploid population size in location $z$ at the time,
so that $1/(2 N_e(z,t))$ is the rate of coalescence there.
Then, we compute the mean coalescence rate as
$$  r(t)  = \frac{1}{p(t)} \sum_z \frac{p(z,t)}{2N_e(z,t)} . $$
This follows because if we have $m$ diploid samples, and hence $\binom{2m}{2}$ lineages,
the expected number of coalescences in location $z$ between times $t$ and $t+dt$ ago is
$$
\binom{2m}{2} p(z,t) \frac{ dt }{ 2 N_e(z,t) },
$$
and the expected number of pairs of uncoalesced lineages at that time is
$$
\binom{2m}{2} p(t) .
$$
The expression for $r(t)$ is a ratio of these two quantities;
to obtain it we need to compute $p(t)$ and $p(z,t)$.
This is relatively straightforward
using the general theory of Markov chains \citep[e.g,.][]{kemeny2012denumerable},
and is implemented in \texttt{msprime}.

Note that since these quantities are \emph{per pair of lineages},
this definition depends on the locations of the samples.
The coalescence rate also has the intuitive interpretation that
it is the average between-lineage coalescence rate,
averaged over where uncoalesced lineages might be.
Since the local coalescence rate is the inverse of the population size,
$1/r(t)$ (as shown for instance in \autoref{fig:n_t_ragsdale})
is a weighted harmonic mean of the census sizes of the different populations present at that time.
This is as expected: suppose that we have two populations, one big and one small,
connected by migration.
If all our samples are from the big population,
the number of recent coalescences should be small, reflecting the large population size,
while in the long run, the coalescence rate approaches an intermediate rate.
On the other hand, more recent coalescences are expected
if all samples are from the small population,
A method that fits a single, time-varying population size to the data
might be expected to find a population size trajectory
to match these time-varying rates of coalescence.

We use the same computations to analytically compute \emph{mean coalescence times}:
since for any nonnegative random variable $T$, the mean value is
$\mathbb{E}[T] = \int_0^\infty \mathbb{P}\{T > t\} dt$,
we can obtain the mean coalescence time as
$$
\int_0^\infty p(t) dt ,
$$
where $p(t)$ is defined above.

The coalescence rate trajectories can be computed from a model in \texttt{msprime}
using the \texttt{coalescence\_rate\_trajectory} method of the Demography Debugger class,
which can be obtained from a \stdpopsim model using the
\texttt{model.get\_demography\_debugger()} method.

\end{document}