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paper/index.qmd

+---
+title: 'SelNeTime: a python package inferring effective population size and selection intensity from genomic time series data'
+tags:
+  - Python
+  - population genetics
+  - genomic time series
+  - Hidden Markov Models
+  - Beta with Spikes distribution
+authors:
+  - name: Mathieu Uhl
+    orcid: 0009-0005-1635-8271
+    affil-id: 1,3
+    affiliation: 
+      - ref: 1
+      - ref: 3
+       # (Multiple affiliations must be quoted)
+  - name: 
+      given: Miguel 
+      family: Navascués
+      dropping-particle: de
+    orcid: 0000-0001-8342-6047
+    affil-id: 1
+    #equal-contrib: true # (This is how you can denote equal contributions between multiple authors)
+    affiliation: 
+      - ref: 1
+  - name: Simon Boitard
+    orcid: 0000-0002-2499-5052
+    equal-contrib: true
+    corresponding: true # (This is how to denote the corresponding author)
+    affil-id: 1
+    affiliation: 
+      - ref: 1
+  - name: Bertrand Servin
+    orcid: 0000-0001-5141-0913
+    equal-contrib: true
+    affil-id: 2
+    affiliation: 
+      - ref: 2
+affiliations:
+ - name: CBGP, Université de Montpellier, CIRAD, INRAE, IRD, Institut Agro Montpellier, Montpellier, France
+   id: 1
+ - name: GenPhySE, Université de Toulouse, INRAE, INPT, Castanet-Tolosan, France
+   id: 2
+ - name: CEFE, CNRS, Montpellier, France
+   id: 3
+date: today
+bibliography: paper.bib
+csl: pci.csl
+---
+
+# Summary
+
+Genomic samples collected from a single population over several
+generations provide direct access to the genetic diversity changes
+occurring within a specific time period. This provides information
+about both demographic and adaptive processes acting on the population
+during that period. A common approach to analyze such data is to model
+observed allele counts in finite samples using a Hidden Markov model
+(HMM) where hidden states are true allele frequencies over time
+(/i.e./ a trajectory). The HMM framework allows to compute the full
+likelihood of the data, while accounting both for the stochastic
+evolution of population allele frequencies along time and for the
+noise arising from sampling a limited number of individuals at
+possibly spread out generations. Several such HMM methods have been
+proposed so far, differing mainly in the way they model the transition
+probabilities of the Markov chain. Following @par2019, we consider
+here the Beta with Spikes approximation, which avoids the
+computational issues associated to the Wright-Fisher model while still
+including fixation probabilities, in contrast to other standard
+approximations of this model like the Gaussian or Beta
+distributions. To facilitate the analysis and exploitation of genomic
+time series data, we present an improved version of @par2019 's
+approach, denoted SelNeTime, whose computation time is drastically
+reduced and which accurately estimates effective population size in
+addition to the selection intensity at each locus. This method is
+implemented in a user friendly python package, which also allows to
+easily simulate genomic time series data under a user-defined evolutionary
+model and sampling design.
+                                                                                   
+# Introduction
+
+One limitation of most population genetic analyses is to use data from
+a single time point, the present. The genetic diversity observed in
+this single sample is very informative about past population history,
+but it integrates the effect of many evolutionary processes acting on
+a large time-scale, which may be hard to clearly separate. For
+instance, selection constraints acting on a given locus often
+fluctuate along time as a result of environmental changes, and such
+complex and changing processes are difficult to reconstruct from the
+observation of the resulting genetic diversity at a single recent time
+point. In contrast, the study of samples collected at several
+different generations allows to directly measure changes in the
+genetic diversity of populations, informing about the evolutionary
+processes acting on that specific time period. Such genomic time
+series arise in different contexts and time scales, such as the
+monitoring of wild or domestic populations [@boi2023], experimental
+evolution [@lon2015] or ancient DNA [@fag2019] studies.
+
+A common approach to analyze genomic time series data is to model
+observed allele frequencies using a Hidden Markov model (HMM) ; this
+approach allows computing the full likelihood of the data, while
+accounting both for the stochastic evolution of population allele
+frequencies along time, under the combined effects of demography and
+selection, and for the noise arising from sampling a limited number of
+individuals at different generations [@bol2008]. Several such HMM
+methods have been proposed so far, differing mainly in the way they
+model the transition probabilities of the Markov chain. Indeed, while
+the Wright-Fisher (WF) model appears as a natural choice to model
+these transitions, it leads to transition matrices whose dimension is
+equal to the population size so it is limited in practice to small
+populations. Among the various approximations (e.g. Gaussian, Beta)
+that have been proposed to overcome this issue, the Beta with Spikes
+(BwS) distribution [@tat2015] was pointed out by @par2019 as
+particularly promising, because it combines a continuous (Beta)
+density with discrete fixation probabilities in 0 and 1. These
+features allow accurate estimations for a large range of demographic
+and selective parameters [@par2019], while previous methods were
+either impossible to run in large populations or inaccurate for some
+parts of the parameter space (small populations, strong selection).
+       
+However, the current implementation of this method, compareHMM
+[@compareHMM], considers a fixed effective population size provided by
+the user, assuming that this parameter is known or has been previously
+estimated from another software. Besides, it was initially implemented
+as a test tool comparing different approximations of the Wright-Fisher
+model in the context of genomic time series analyses, not as a real
+package to be used by a large community. In particular, it was not
+optimized to analyze very large data sets such as those typically
+found in the genomics era.
+
+With the objective of improving on these several limitations, the
+SelNeTime package provides a user friendly and computationaly
+efficient implementation of @par2019's method, allowing to estimate
+effective population size and selection from genomic time series and
+to simulate such data in order to easily test the expected performance
+of the approach under custom sampling designs.
+                                                  
+# Important features
+
+## Estimation of effective population size
+
+Given an observed temporal sequence of sampled allele counts
+$\mathbf{Y}_i$ at a given locus $i$, the HMM approach described in
+@par2019 can evaluate the (exact) likelihood $P(\mathbf{Y}_i|N,s_i)$
+for any values of the effective population size $N$ and of the
+selection intensity $s_i$. As indicated in the notations, $s_i$ is a
+locus specific parameter, because selective contraints typically
+differ between loci. In contrast, $N$ reflects the demography of the
+population so a single value applies to the whole genome. Considering
+$p$ loci assumed to be independent (which almost holds in practice if
+these loci are taken sufficiently far away from each other) and
+neglecting in first approximation the effect of selection at each of
+these loci, SelNeTime computes the likelihood of $N$ as $$
+P(\mathbf{Y}_1,\dots,\mathbf{Y}_p|N)=\prod_{i=1}^p
+P(\mathbf{Y}_i|N,s_i=0) $$ and finds the Maximum Likelihood Estimator
+(MLE) of $N$, i.e. the value of $N$ maximizing the above formula.
+
+
+![Estimation of the effective population size for 100 datasets, using SelNeTime or NB [@nb]. Each dataset includes 1000 loci and 10 sampling times $t=1\dots10$ (red) or $t=1,10,\dots,100$ (blue). Allele frequency trajectories at each locus were simulated under the Wright-Fisher model with $N=100$, $s=0$ and an initial allele frequency $\pi_0=0.5$. 30 alleles were observed at each sampled generation.](figures/Figure1.png){#fig-nb}
+
+Using computer simulations, we checked that this approach provides
+accurate estimates of $N$, as illustrated in @fig-nb. As a comparison,
+we estimated $N$ for the same data with the method of @hui2015,
+implemented in the NB package [@nb], which is based on a continuous
+Beta approximation of allele frequency transitions. Interestingly,
+both methods performed equally well for short sampling time intervals
+($dt=1$) but the accuracy of NB was reduced for larger time intervals
+($dt=10$); this is consistent with the higher probability of fixation
+events, which are not accounted in the method of @hui2015, when $dt$
+increases, illustrating the advantage of the BwS approximation.
+
+
+## Estimation of selection intensity at a given locus
+
+![Estimation of the selection intensity for 1000 loci using SelNeTime. Allele frequency trajectories at each locus were simulated under the Wright-Fisher model with $N=100$, $s=0$ (purple) or $s=0.1$ (yellow) and an initial allele frequency $\pi_0=0.5$. 30 alleles were observed at 10 sampling times $t=1\dots10$.](figures/Figure2.png){#fig-sel}
+
+Once the effective population size $N$ has been estimated, SelNeTime
+estimates the selection intensity at each locus $i$ using the same
+approach as @par2019, i.e. optimizing $P(\mathbf{Y}_i|N,s_i)$ over
+different values of $s_i$ in order to find the MLE of this
+parameter. This leads to unbiased estimations of the selection
+intensity (@fig-sel), as already explored in details by @par2019 for a
+large range of parameter values and various sampling designs.
+
+
+## Improved computation time
+
+Another important improvement of SelNeTime compared to the inital
+implementation of the BwS approach in @compareHMM concerns the
+computation time, which was reduced using two main actions. First, the
+continuous density of the distribution was discretized when building
+the transition matrices (with a controlled bin width), allowing this
+continuous part of the distribution to be analyzed exactly in the same
+way as the discrete probability mass points in 0 and 1. Second, matrix
+computations performed at each iteration of the HMM forward algorithm
+were vectorized, taking advantage of the Numpy library
+[@harris2020array] to efficiently handle different loci or parameter
+values in a single operation.
+
+As an illustration, @tbl-computetime shows the computation time
+required to analyze a time series of 10 samples of 30 alleles each,
+with a sampling time interval of $dt=10$ generations and a number of
+loci varying from 100 to 10000.
+
+| Nb. loci   | @compareHMM, estimation of $s$ | SelNeTime, estimation of $s$ | SelNeTime, estimation of $N$ |
+|:-------:|:--------:|:----------:|:----------:|
+| 100     |    39.27s    |      28.7s    | 6.12s |
+| 1000     |     360.01s   |    36.21s      | 14.23s |
+| 10000     |   3530.47s     |      96.95s    | 87.83s |
+
+: Computation times of CompareHMM and SelNeTime {#tbl-computetime}
+
+For 100 loci, SelNeTime runs in approximately the same time as
+@compareHMM, but the relative performance of SelNeTime improves
+quickly when the number of loci increases. Actually, the time to
+compute all transition matrices, which is the first step of both
+SelNeTime and @compareHMM analyses, is independent of the number of
+loci (but it depends on the sampling design). This part of the
+analysis took approximately 28s for SelNeTime in this example, which
+was of the same order as @compareHMM, but after this the time spent
+per locus was far lower with SelNeTime (approx. 0.007s) than with
+@compareHMM (approx. 0.35s).
+
+## Comparison with the Wright-Fisher model
+
+While SelNeTime computes by default HMM transitions using the BwS
+distribution, it might occasionaly be useful to compute them based on
+the WF model, in particular to evaluate the impact of the BwS
+approximation on inference quality. Although this is numerically
+possible only for small population sizes (not more than a few
+hundreds), we included one option to do this in the program. For a
+true population size of $N=100$, we could for instance check that
+using the BwS or the WF model had very limited impact on the inference
+of either effective population size or selection (@fig-wf). A more
+detailed comparison between the estimations obtained by the BwS and WF
+distributions can be found in @par2019, but only for selection
+inference.
+
+![Comparing of SelNeTime estimations based on the Beta with Spikes and the Wright-Fisher model. Left: MLE of effective population size for 100 datasets of 1000 loci each. (B) MLE of $s$ for 1000 loci. The two panels show the estimations obtained using the WF (x axis) or the BwS (y axis) model on the same data, which were simulated under the Wright-Fisher model with $N=100$ $s=0$ (Left and Right, purple) or $s=0.1$ (Right, yellow) and initial allele frequencies $\pi_0=0.5$. 30 alleles were observed at 10 sampling times $t=1,10\dots100$.](figures/Figure3.png){#fig-wf}
+
+## Simulation of datasets
+   
+As already observed in @par2019 and several other studies on the
+subject, the estimation accuracy of effective population size or
+selection depends on both the true evolutionary parameters of the
+population under study ($N$ and $s$) and the sampling design of
+observed data (number and size of observed samples, time intervals
+between samples). In order to help researchers plan future
+experiments, or interprete the results obtained for a specific
+dataset, SelNeTime includes a simulator of observed allele frequency
+trajectories. Simulations can easily be tunned using either a yaml
+file or command line options and output data at a format that can
+directly be analyzed with other SelNeTime commands. Notably, this
+simulator was used to generate the data underlying the figures and
+tables of this study.
+
+# Acknowledgments
+
+This study has recieved the support of the Occitanie Regional Council’s program 'Key challenge BiodivOc'.
+We are also grateful to the genotoul bioinformatics platform Toulouse Occitanie
+(Bioinfo Genotoul, https://doi.org/10.15454/1.5572369328961167E12) for providing computing resources.
+
+# References
+                                                                                                             
+::: {#refs}
+:::                 

paper/paper.bib

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+ title         = {Array programming with {NumPy}},
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+                 Travis E. Oliphant},
+ year          = {2020},
+ month         = sep,
+ journal       = {Nature},
+ volume        = {585},
+ number        = {7825},
+ pages         = {357--362},
+ doi           = {10.1038/s41586-020-2649-2},
+ publisher     = {Springer Science and Business Media {LLC}},
+ url           = {https://doi.org/10.1038/s41586-020-2649-2}
+}
+
+
+@article{boi2021,
+  title={Gene banks as reservoirs to detect recent selection: the example of the Asturiana de los Valles bovine breed},
+  author={Boitard, Simon and Paris, Cyriel and Sevane, Natalia and Servin, Bertrand and Bazi-Kabbaj, Kenza and Dunner, Susana},
+  journal={Frontiers in genetics},
+  volume={12},
+  year={2021},
+  publisher={Frontiers Media SA}
+}
+
+@article{boi2023,
+  title={Whole-genome sequencing of cryopreserved resources from French Large White pigs at two distinct sampling times reveals strong signatures of convergent and divergent selection between the dam and sire lines},
+  author={Boitard, Simon and Liaubet, Laurence and Paris, Cyriel and F{\`e}ve, Katia and Dehais, Patrice and Bouquet, Alban and Riquet, Juliette and Mercat, Marie-Jos{\'e}},
+  journal={Genetics Selection Evolution},
+  volume={55},
+  number={1},
+  pages={13},
+  year={2023},
+  publisher={Springer}
+}
+
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+	author = "Bollback, Jonathan P. and York, Thomas L. and Nielsen, Rasmus",
+	journal = "Genetics",
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+	x-color = "#009966",
+	year = "2008"
+}
+
+
+@misc{compareHMM,
+  author = {Paris, Cyriel and Servin, Bertrand and Boitard, Simon},
+  title = {compareHMM},
+  year = {2019},
+  publisher = {GitHub},
+  journal = {GitHub repository},
+  url = {https://github.com/CyrielParis/compareHMM}
+}
+
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+	issn = "0092-8674",
+	journal = "Cell",
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+	pages = "1419--1435.e31",
+	publisher = "Elsevier",
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+	volume = "177",
+	year = "2019"
+}
+
+@article{hui2015,
+	author = "Hui, Tin-Yu J. and Burt, Austin",
+	issn = "0016-6731",
+	journal = "Genetics",
+	number = "1",
+	pages = "285--293",
+	publisher = "Genetics",
+	title = "{Estimating Effective Population Size from Temporally Spaced Samples with a Novel, Efficient Maximum-Likelihood Algorithm}",
+	volume = "200",
+	year = "2015"
+}
+
+@article{lon2015,
+  title={Elucidating the molecular architecture of adaptation via evolve and resequence experiments},
+  author={Long, Anthony and Liti, Gianni and Luptak, Andrej and Tenaillon, Olivier},
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+  number={10},
+  pages={567--582},
+  year={2015},
+  publisher={Nature Publishing Group UK London}
+}
+
+@article{mal2015,
+	author = "Malaspinas, Anna-Sapfo",
+	journal = "Molecular Ecology",
+	month = nov,
+	number = "1",
+	pages = "24--41",
+	title = "{Methods to characterize selective sweeps using time serial samples: an ancient DNA perspective}",
+	volume = "25",
+	year = "2015"
+}
+
+@misc{nb,
+  author = {"Hui, Tin-Yu J."},
+  title = {NB: Maximum Likelihood method in estimating effective population size from genetic data},
+  year = {2014},
+  publisher = {CRAN},
+  journal = {CRAN repository},
+  url = {https://cran.r-project.org/web/packages/NB}
+}
+
+@article{par2019,
+  title={Inference of selection from genetic time series using various parametric approximations to the Wright-Fisher model},
+  author={Paris, Cyriel and Servin, Bertrand and Boitard, Simon},
+  journal={G3: Genes, Genomes, Genetics},
+  volume={9},
+  number={12},
+  pages={4073--4086},
+  year={2019},
+  publisher={Oxford University Press}
+}
+
+@article{tat2015,
+	author = "Tataru, Paula and Bataillon, Thomas and Hobolth, Asger",
+	doi = "10.1534/genetics.115.179606",
+	eprint = "https://www.genetics.org/content/201/3/1133.full.pdf",
+	issn = "0016-6731",
+	journal = "Genetics",
+	number = "3",
+	pages = "1133--1141",
+	publisher = "Genetics",
+	title = "{Inference under a wright-fisher model using an accurate beta approximation}",
+	url = "https://www.genetics.org/content/201/3/1133",
+	volume = "201",
+	year = 2015
+}
+

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+      <substitute>
+        <names variable="editor"/>
+        <names variable="translator"/>
+        <choose>
+          <if type="bill book graphic legal_case legislation motion_picture report song" match="any">
+            <text variable="title" form="short" font-style="italic"/>
+          </if>
+          <else>
+            <text variable="title" form="short" quotes="true"/>
+          </else>
+        </choose>
+      </substitute>
+    </names>
+  </macro>
+  <macro name="title">
+    <choose>
+      <if type="bill book graphic legal_case legislation motion_picture report song" match="any">
+        <text variable="title" font-style="italic"/>
+      </if>
+      <else>
+        <text variable="title"/>
+      </else>
+    </choose>
+  </macro>
+  <macro name="publisher">
+    <group delimiter=", ">
+      <text variable="publisher"/>
+      <text variable="publisher-place"/>
+    </group>
+  </macro>
+  <citation collapse="year-suffix" et-al-min="3" et-al-use-first="1" et-al-subsequent-min="3" et-al-subsequent-use-first="1" disambiguate-add-year-suffix="true" disambiguate-add-names="true">
+    <sort>
+      <key variable="issued"/>
+      <key variable="author"/>
+    </sort>
+    <layout prefix="(" suffix=")" delimiter="; ">
+      <group delimiter=", ">
+        <text macro="author-short"/>
+        <date variable="issued">
+          <date-part name="year"/>
+        </date>
+        <group>
+          <label variable="locator" suffix="." form="short" strip-periods="true"/>
+          <text variable="locator" prefix=" "/>
+        </group>
+      </group>
+    </layout>
+  </citation>
+  <bibliography hanging-indent="true" et-al-min="300" et-al-use-first="3" entry-spacing="0" line-spacing="2">
+    <sort>
+      <key macro="author-short"/>
+      <key variable="issued"/>
+    </sort>
+    <layout>
+      <text macro="author" suffix=" "/>
+      <date variable="issued" prefix="(" suffix=")">
+        <date-part name="year"/>
+      </date>
+      <choose>
+        <if type="bill book graphic legal_case legislation motion_picture report song" match="any">
+          <group suffix=".">
+            <text macro="title" prefix=" "/>
+            <text macro="editor-translator" prefix=" "/>
+          </group>
+          <text prefix=" " suffix="." macro="publisher"/>
+        </if>
+        <else-if type="thesis" match="any">
+          <group suffix=".">
+            <text macro="title" prefix=" " suffix="."/>
+            <text variable="genre" prefix=" " suffix=" Thesis."/>
+            <text prefix=" " macro="publisher"/>
+          </group>
+        </else-if>
+        <else-if type="chapter paper-conference" match="any">
+          <text macro="title" prefix=" "/>
+          <group prefix=".">
+            <group prefix=" In:" suffix="">
+              <text variable="container-title" font-style="italic" prefix=" " suffix=" "/>
+              <text variable="collection-title" prefix=" " suffix="."/>
+              <names variable="editor translator" prefix=" (" delimiter=", " suffix=")">
+                <label form="short" text-case="lowercase" suffix=" " strip-periods="true"/>
+                <name name-as-sort-order="all" sort-separator=" " initialize-with="" delimiter=", " delimiter-precedes-last="never"/>
+              </names>
+              <group suffix=".">
+                <label variable="page" form="short" prefix=", " suffix=". " strip-periods="true"/>
+                <text variable="page" suffix=""/>
+                <text macro="publisher" prefix=". " suffix="."/>
+              </group>
+            </group>
+          </group>
+        </else-if>
+        <else>
+          <group suffix=".">
+            <text macro="title" prefix=" "/>
+            <text macro="editor-translator" prefix=" "/>
+          </group>
+          <group prefix=" " suffix=".">
+            <text variable="container-title" font-style="italic"/>
+            <group prefix=", ">
+              <text variable="volume" font-weight="bold"/>
+            </group>
+            <text variable="page" prefix=", "/>
+          </group>
+        </else>
+      </choose>
+    <text variable="DOI" prefix=". https://doi.org/"/>
+    </layout>
+  </bibliography>
+</style>
\ No newline at end of file

paper/title.tex

+$if(title)$
+\title{$title$$if(thanks)$\thanks{$thanks$}$endif$}
+$endif$
+
+$if(subtitle)$
+\subtitle{$subtitle$}
+$endif$
+
+$for(by-author)$
+\author$if(it.metadata.affil-id)$[$it.metadata.affil-id$]$endif${$it.name.literal$}
+$endfor$
+
+$if(by-affiliation)$
+$for(by-affiliation)$
+\affil[$it.id$]{$it.name$}
+$endfor$
+$endif$
+
+
+\date{$date$}