LEVERAGING LARGE DATASETS TO INVESTIGATE
THE DISTRIBUTION OF FITNESS EFFECTS IN PLANTS

By

Miles David Roberts

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Genetics and Genome Sciences – Doctor of Philosophy

2025

ABSTRACT

Mutations—spontaneous changes in DNA sequences—are fundamental to evolution, and con-

tribute to everything from disease susceptibility in people to agriculture. Population geneticists are

tasked with understanding mutations - historically categorizing them as either beneficial, harmful,

or neutral based on how they affect fitness. However, our understanding of these categories is of-

ten limited by incomplete data and an inability to travel back in time to track mutations. Over four

chapters, I will explore how large datasets can enhance our understanding of these three funda-

mental mutation categories. First, I will use 27 terabases of gene expression data from 300 studies

to investigate whether rarely expressed genes accumulate harmful mutations at higher rates than

constitutively expressed genes. Next, I will leverage about 205 terabases of DNA-sequencing data

and use k-mers (DNA subsequences) to measure neutral variation in 112 natural plant species. The

results suggest that current methods for estimating diversity reliant on reference genomes underes-

timate genetic variation. The following chapter expands upon the idea of using k-mers to measure

genetic diversity, numerically and analytically exploring the relationship between k-mer diversity

and nucleotide diversity. Overall, k-mer diversity scales linearly with nucleotide diversity and we

showcase the use of bloom filters to decrease the memory burden of the k-mer diversity calcu-

lations. Finally, I will use 250,000 evolutionary simulations to train machine learning models to

infer the time it took for a fixed beneficial mutation to spread. Overall, large datasets like these

hold many opportunities to revisit old biological questions and further our understanding of muta-

tion trajectories.

To David, Cheree, Blake, and Paige Roberts

iii

ACKNOWLEDGEMENTS

I am infinitely appreciative for the guidance and support of my advisor, Emily Josephs. Emily’s

patience, even in the face of my constant stubbornness, and her promotion of curiosity greatly

shaped how I view research. This dissertation also greatly benefited from the input of my com-

mittee: Bob Vanburen, Shinhan Shiu, and Jeff Connor. I want to give a special thanks to Addie

Thompson for being the outside evaluator for my comprehensive exam and for later stepping into

my committee for my dissertation defense. I also want to thank Olivia Davis and Robert Williamson

for being wonderful collaborators. We will see their awesome work in chapter 3.

I deeply appreciate the members of the Josephs lab, past and current, who have given me invalu-

able advice and have (usually) been willing to laugh at my puns. These folks are: Husain Agha,

Sophie Buysse, Nate Catlin, Derek Denney, Asia Hightower, Daniela Palmer, Rebecca Panko,

Gabbie Sandstedt, Mia Stevens, Chrissy Miller, Adrian Platts, Magie Williams, and Maya Wilson

Brown. I was also lucky to have many amazing friends that helped keep my sprits high. In no

particular order, these folks are: Laura Ford, Nick Johnson, Max Harman, Luke Strickland, Bran-

don Webster, Kim Fisher, Brandon Beal, John Botsford, Robin Waterman, and Carolyn Graham.

Thank you all.

I also want to thank MSU’s Council of Graduate Students and their dozens of representatives

across MSU for their work in advocating for graduate and professional student needs and for giving

me the opportunity to be their President. It was a great honor to represent my peers in that capacity

and the experience shaped my understanding of leadership. Thank you especially to Deanne Arking

and Allyn Shaw for helping me immensely with these efforts.

I want to acknowledge that it was my undergraduate genetics teacher William Cushwa that

encouraged me to continue studying genetics and to engage in undergraduate research, which were

both crucial to me pursuing a graduate degree.

I did undergraduate research in Dr. Stephanie

Porter’s lab at Washington State University in Vancouver, Washington and I received incredibly

valuable advice and mentorship that I continue to lean on to this day. I also want to acknowledge

mentorship from the former post-docs Emily Helliwell and Camile Wendlandt, as well as former

iv

graduate students Angeliqua Montoya, Zoie Lopez, and Niall Millar that were willing to train and

mentor me during this time and motivated me to go to graduate school.

The work in this dissertation, as with all scientific works, leans heavily on the previous efforts

of other scientists. I want to acknowledge that in chapters 1 and 2, I make extensive use of public

databases with contributions from hundreds of scientists across the world. It is impossible to list all

of the contributors here, but I am grateful for their work and for making their data public. Without

their effort none of this would have been possible.

I am also grateful for the anonymous peer

reviewers that have commented on chapters 1-3.

I would like to thank the Plant Biology Department and the Ecology, Evolution, and Behavior

Program at MSU for providing a welcoming environment where I could always participate even

though I had technically no affiliation with either of them. My home program of Genetics and

Genome Sciences was also always supportive and flexible in allowing me to pursue my personal

educational goals.

I would like to thank the Biomolecular Sciences Program for supporting the 1st year of my PhD,

the National Science Foundation and the National Institutes of Health for supporting the 2nd, 3rd,

and 4th years of my PhD. I would also like to thank MSU’s Institute for Cyber-Enabled Research

for providing the computing power and a fellowship to learn cloud-based computing approaches.

To my family, David, Cheree, Blake, and Paige Roberts, your support is the reason why I got

to pursue an education. I do not know where I would be without it. I love you all very much. To

my partner, Riley Pizza, thank you for your endless support. You are the single greatest thing that’s

happened to me in graduate school and I cannot wait to see where our journey takes us.

- Miles

v

PREFACE

This dissertation is my original work. Chapters 1 and 3 are published in peer-reviewed journals

under a CC-BY license, allowing unlimited sharing and reproduction as long as attribution is given.

Chapter 2 is accepted for publication at Evolution Letters, but the preprint version is available

at biorxiv under a CC-BY license. Chapter 4 is unpublished. I was responsible for all of data

collection, data analysis, and writing in Chapters 1, 2, and 4 (with guidance from my advisor and

committee). Chapter 3 was co-first authored by myself and Olivia Davis - a former undergraduate

(now graduate) of Rose-Hulman Institute of Technology in Terre Haute, Indiana. Olivia carried out

the simulations depicted in Figures 3.3 - 3.5, D1, and D2. I performed the simulations depicted in

Figure 3.1, made Figure 3.2, and derived equations 3.7, 3.8, and 3.15. Olivia also wrote the text in

the section Testing the efficacy of k-mer measures of variation, but all other text was written by

me.

vi

TABLE OF CONTENTS

INTRODUCTION .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 1: WEAKER SELECTION ON GENES WITH TREATMENT SPECIFIC
EXPRESSION CONSISTENT WITH A LIMIT ON PLASTICITY EVOLUTION
IN ARABIDOPSIS THALIANA . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

CHAPTER 2: k-MER-BASED DIVERSITY SCALES WITH POPULATION SIZE PROX-

IES MORE THAN NUCLEOTIDE DIVERSITY IN A META-ANALYSIS OF 98
PLANT SPECIES . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

.

.

CHAPTER 3: k-MER-BASED APPROACHES TO BRIDGING PANGENOMICS AND

POPULATION GENETICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

CHAPTER 4: SUMMARY STATISTICS COMPARABLE TO CONVOLUTIONAL

NEURAL NETWORKS FOR INFERRING TIMES TO FIXATION . . . . . . . . 90

BIBLIOGRAPHY .

. .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

APPENDIX A: SUPPLEMENTAL FIGURES FOR CHAPTER 1 . . . . . . . . . . . . 139

APPENDIX B: SUPPLEMENTAL METHODS FOR CHAPTER 2 . . . . . . . . . . . . 178

APPENDIX C: SUPPLEMENTAL FIGURES FOR CHAPTER 2 . . . . . . . . . . . . 179

APPENDIX D: SUPPLEMENTAL FIGURES FOR CHAPTER 3 . . . . . . . . . . . . 205

APPENDIX E: SUPPLEMENTAL FIGURES FOR CHAPTER 4 . . . . . . . . . . . . . 207

vii

INTRODUCTION

Population geneticists study genetic variation within and between populations. To understand

variation, we need to understand the fate of new mutations. What determines whether a mutation

spreads through a population or goes extinct? There are three frontiers that are currently revolu-

tionizing our understanding of this question: (1) the collection of data across many environments,

(2) the assembly of multiple genomes per population (i.e. pangenomics), and (3) machine learning

(Hahn, 2018). In the course of this dissertation, we will familiarize ourselves with each of these

areas, but to prepare for the journey, we will need a mental model as a guide.

Let’s imagine the amount of genetic diversity in a population as the level of water in a pool. In

this analogy, the process of mutation is the faucet or hose that adds water into the pool. If this were

the only dynamic, then our pool would fill up forever. Thus, our pool must have some drains. One

important drain is natural selection, which is a process where mutations that cause heritable, delete-

rious changes in survival or reproduction will not spread through a population and may eventually

go extinct (i.e. purifying selection). On the other hand, mutations that cause heritable, beneficial

changes to survival and reproduction will spread and may eventually be fixed (i.e. positive se-

lection). If we imagine a world where all mutations are deleterious then we would expect natural

selection to constantly remove mutations from our population. However, this expectation clearly

does not match reality because essentially all natural populations have genetic variation (Leffler

et al., 2012). One way to modify our expectation is to recognize that natural selection is not an

omnipotent force - the pool drain only opens under certain circumstances.

One important limit to natural selection is that it can only act when a mutation is affecting fit-

ness. If a mutation occurs in a non-expressed gene, for instance, then such a mutation may not

affect fitness and will be invisible to natural selection. This masking of deleterious mutations can

theoretically happen in any gene that is only expressed at certain times or in certain environments

(Van Dyken and Wade, 2010). But on the other hand, natural selection often favors conditional ex-

pression (Snell-Rood et al., 2010). Many species of plants have evolved specific gene expression

responses for specific stimuli, including mechanical stress (Braam and Davis, 1990), light stress

1

(Mishra et al., 2012), and heat stress (Zhang et al., 2017b). Thus, a trade-off arises between the

added benefit of conditional expression vs the weakened ability to purify genes of deleterious muta-

tions. Though the theory behind this relationship is well-understood (Van Dyken and Wade, 2010),

there are no empirical investigations of the relationship between environment-specific expression

and deleterious variation. This is in part because gene expression could not be measured across

many environments easily until relatively recently. Chapter 1 leverages publicly available gene

expression data across 200 different environments in Arabidopsis thaliana to test whether genes

with more environment-specific expression have more deleterious variation.

What if we assume most mutations are neutral? This idea is the foundation of Neutral The-

ory (Kimura, 1983). Under this framework, the amount of variation in a population is simply a

product of how many individuals are in the population and the rate of mutation. In other words,

more individuals replicating DNA to produce offspring means more opportunities for mutations to

occur in DNA. This is a straightforward expectation that is anchored in the foundational works of

population genetics and evolutionary biology. Charles Darwin observed that “wide ranging, much

diffused and common species vary most” (Darwin (1859), Chapter II) and R. A. Fisher supported

this observation mathematically as “a numerous species, with the same frequency of mutation, will

maintain a higher variability than will a less numerous species” (Fisher, 1923). However, when sci-

entists empirically estimate the scaling between population size and genetic diversity, the result is

much weaker than expected (Lewontin, 1974; Leffler et al., 2012; Buffalo, 2021). There are many

biological mechanisms that contribute to this paradoxical result (Charlesworth and Jensen, 2022).

However, one underexplored explanation is whether our methods of measuring genetic diversity

simply do not capture all of the variation present. Chapters 2 and 3 will investigate whether the

ubiquitous practice of using a reference genome to estimate diversity contributes to this paradox

and how patterns in k-mers (DNA subsequences) provide a potential solution.

Finally, what about beneficial mutations? Positive selection can also be thought of as a drain

in our pool analogy because fixation of beneficial alleles removes diversity around the site of the

beneficial mutaiton - a signature called a selective sweep (Whitehouse and Schrider, 2023). The

2

amount of diversity that positive selection removes is proportional to the time it takes for the mu-

tation to fix (Coop, 2020). Shorter times to fixation result in greater loss of diversity because there

is less time for mutation and recombination to break up the sweep. After fixation, mutation and

recombination will gradually reintroduce variation at the site of the sweep over time. This suggests

that the times to fixation and age of a sweep are encoded in patterns of genomic diversity around

the sweep site. However, times to fixation rarely estimated in real data, despite being a core quan-

tity in studies of neutral theory and selective sweeps (Charlesworth, 2020; Zhao et al., 2013). In

chapter 4, we will use machine learning to try to estimate times to fixation using a snapshot of a

contemporary population.

3

CHAPTER 1: WEAKER SELECTION ON GENES WITH TREATMENT SPECIFIC
EXPRESSION CONSISTENT WITH A LIMIT ON PLASTICITY EVOLUTION IN
ARABIDOPSIS THALIANA1

Abstract

Differential gene expression between environments often underlies phenotypic plasticity. How-

ever, environment-specific expression patterns are hypothesized to relax selection on genes, and

thus limit plasticity evolution. We collated over 27 terabases of RNA-sequencing data on Ara-

bidopsis thaliana from over 300 peer-reviewed studies and 200 treatment conditions to investigate

this hypothesis. Consistent with relaxed selection, genes with more treatment-specific expression

have higher levels of nucleotide diversity and divergence at nonsynonymous sites but lack stronger

signals of positive selection. This result persisted even after controlling for expression level, gene

length, GC content, the tissue specificity of expression, and technical variation between studies.

Overall, our investigation supports the existence of a hypothesized trade-off between the environ-

ment specificity of a gene’s expression and the strength of selection on said gene in A. thaliana.

Future studies should leverage multiple genome-scale datasets to tease apart the contributions of

many variables in limiting plasticity evolution.

Introduction

Organisms must cope with ever-changing environmental conditions to survive and reproduce.

If these changes in condition cannot be avoided or escaped, phenotypes that respond to environ-

mental variation through phenotypic plasticity may be adaptive. For example, under low light, the

same Arabidopsis thaliana genotype will produce more or larger leaves to capture more energy for

photosynthesis (Pigliucci and Kolodynska, 2002). Plastic responses are partly controlled through

differential gene expression between environments (Scheiner, 1993; Schlichting and Smith, 2002).

Understanding the evolution of these condition-specific expression patterns could help reconcile

the diversity of plastic responses observed in nature and engineer organisms to overcome environ-

mental challenges.

1This chaper is published at the following DOI under a Creative Commons CC-BY License: https://doi.org/10.
1093/genetics/iyad074. This license permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited. Permission to reuse this article is not required.

4

However, not all organisms can respond plastically to environmental change, so it is crucial

to understand the processes that constrain plasticity (Van Kleunen and Fischer, 2005). These con-

straints are usually characterized as either costs, where plasticity reduces fitness in some way, or

limits to the evolution or maintenance of plasticity (DeWitt et al., 1998). Decades of research has

attempted to measure the costs associated with plasticity (reviewed in Schneider 2022) but studies

often fail to detect costs or find costs that are weak or restricted to certain environments (Van Kle-

unen and Fischer, 2005; Van Buskirk and Steiner, 2009; Auld et al., 2010). Theory also predicts

that there will be strong selection to alleviate costs (Murren et al., 2015). Thus, limits may be more

important than costs in shaping the evolution of plasticity.

Recent work suggests that relaxed selection can limit plasticity evolution (Snell-Rood et al.,

2010; Murren et al., 2015). For instance, one hypothesis posits that genes are often under selection

for environment-specific expression to minimize deleterious pleiotropy (Snell-Rood et al., 2010;

McGuigan et al., 2014; Huber et al., 2017). However, narrowing the range of environments where

a gene is expressed also reduces the opportunity for negative selection to act on deleterious muta-

tions in the gene (Kawecki, 1994; Whitlock, 1996; Van Dyken and Wade, 2010). The accumulation

of deleterious mutations could then cancel out any selective benefits of the environment-specific

expression pattern. Thus, a trade-off arises between a gene’s degree of environment-specific ex-

pression and the strength of negative selection acting on said gene. If we assume that environment-

specific expression generally contributes to phenotypic plasticity, then this trade-off would poten-

tially limit the maintenance of plasticity (Kawecki, 1994; Snell-Rood et al., 2010). Whether such

a trade-off exists has not yet been tested, but the deposition of expression data from hundreds of

experimental treatments across hundreds of labs into public repositories now enables approximat-

ing environment specificity as treatment specificity and linking treatment-specific expression to

the rate of evolution.

One challenge in studying the relationship between treatment specificity and protein evolution

is that many factors influence evolutionary rates (for review, see Rocha 2006; Gaut et al. 2011;

Koonin 2011; Zhang and Yang 2015) and these factors are hard to disentangle. A protein’s expres-

5

sion level is often considered the best predictor of its evolutionary rate (Rocha, 2006) - a result

observed across all domains of life (Zhang and Yang, 2015) and sometimes considered a “law” of

genome evolution (Koonin, 2011). Among multicellular organisms, the degree of tissue specificity

in expression is also generally predictive of evolutionary rates (Duret and Mouchiroud, 2000; Lar-

racuente et al., 2008; Winter et al., 2004; Zhang and Li, 2004; Slotte et al., 2011; Bush et al., 2015;

Mukherjee et al., 2016; Groen et al., 2020; Huang, 2022). Additional factors that also influence

evolutionary rates include exon edge conservation (Bush et al., 2015), mutational bias (Wang et al.,

2004; Ossowski et al., 2010), gene length (Mukherjee et al., 2016), gene age (Moutinho et al.,

2022), GC content (Zhang et al., 2002; Mukherjee et al., 2016), expression stochasticity (Groen

et al., 2020), involvement in general vs specialized metabolism (Mukherjee et al., 2016), iden-

tity as a regulatory or structural gene (Wheeler et al., 2022), recombination rate (Langley et al.,

2012), codon-bias (Betancourt and Presgraves, 2002), mating system (Wright et al., 2002; Glémin,

2007; Payne and Alvarez-Ponce, 2018), gene compactness (Larracuente et al., 2008; Mukherjee

et al., 2016), co-expression or protein-protein interaction network connectivity (Alvarez-Ponce and

Fares, 2012; Masalia et al., 2017; Mähler et al., 2017; Alvarez-Ponce et al., 2017; Josephs et al.,

2017), gene body methylation (Takuno and Gaut, 2012), metabolic flux (Colombo et al., 2014),

protein structure (Lin et al., 2007), essentiallity (Nembaware et al., 2002; Yang et al., 2003; Davis

and Petrov, 2004), and even plant height (Lanfear et al., 2013). This overabundance of possible

explanatory variables suggests that massive genome-scale datasets and careful statistical analysis

are required to tease out the influence of treatment-specific expression on evolutionary rates.

To investigate the influence of treatment-specific expression on evolutionary rates, we com-

piled a dataset of gene expression data across over 200 treatments from over 300 peer-reviewed

studies in A. thaliana. We annotated RNA-sequencing runs from these studies using standardized

ontologies, then processed all of them with the same pipeline. Finally, we combined the result-

ing gene expression matrix with estimates of selection based on within-species polymorphism and

between-species divergence to investigate whether genes with treatment-specific expression were

under weaker negative selection.

6

Methods

RNA-seq run annotation

We amassed an initial set of RNA-seq runs from the Sustech Arabidopsis RNA-seq database

V2 (Zhang et al., 2020) (http://ipf.sustech.edu.cn/pub/athrdb/) excluding any samples not associ-

ated with a publication or lacking a tissue type label. On May 24th, 2022 we also downloaded

all run metadata from the Sequence Read Archive (SRA) returned by the following search term:

(“Arabidopsis thaliana”[Organism] AND “RNA”[Source]) OR (“Arabidopsis thaliana”[Organism]

AND “RNA-Seq”[Strategy]) OR (“Arabidopsis thaliana”[Organism] AND “TRANSCRIPTOM-

IC”[Source]). All SRA runs were linked to their associated publications, if possible, using Entrez.

Any SRA run numbers that we could not link to a PUBMED ID or DOI were omitted. We then man-

ually removed all SRA runs that originated from transgenic, mutant, hybrid, grafted, cell culture,

polyploid, or aneuploid samples based on information in the SRA metadata and associated publi-

cations. Runs from any naturally-occurring A. thaliana accession were included. We also omitted

SRA runs that focused on sequencing non-coding RNA (ncRNA-seq, miRNA-seq, lncRNA-seq,

sRNA-seq, etc.). After applying these criteria, any bioprojects with 8 or fewer SRA run numbers

remaining were also omitted.

All runs were labeled with treatment and tissue type descriptions using the Plant Experimen-

tal Conditions Ontology (PECO) and the Plant Ontology (PO) (Cooper et al., 2018), respectively,

based on information in their associated publications and SRA metadata. In our analysis, control

exposure was defined as long day conditions (12 hrs light exposure or longer, but not constant light)

and growing temperatures in the range of 18° - 26°, inclusive, without explicit application of stress

or nutrient limitation. Warm treatments were defined as 27° or higher, while cold treatments were

defined as 17° or lower. Any studies that did not report both day length and growing temperature

were omitted. Any runs that could not be linked to treatments based on their annotations in the SRA

or Sustech databases were also omitted. Treatment with polyethylene glycol (PEG) was catego-

rized as drought exposure. Samples from plants that were recovering from stress were categorized

according to the growth conditions of the recovery state instead of the stressed state. When appro-

7

priate, we labeled samples with multiple PECO terms. For example, a sample that was subjected to

both heat stress and high light stress would get two PECO terms (one for each stress) and be treated

separately from samples subjected to only heat stress or only light stress. Tissue type labels were

eventually collapsed to the following categories: whole plant, shoot, root, leaf, seed, and a com-

bined category of flower and fruit tissues. The flower and fruit tissue categories were combined

because of their developmental relationship and small size relative to the other categories. In the

end, we had a dataset of 24,101 sequencing runs from 306 published studies.

RNA-seq run processing

All RNA-seq runs were processed using the same workflow to remove the effects of bioin-

formatic processing differences between studies on expression level. First, runs were downloaded

using the SRA toolkit (v2.10.7), but 90 runs were not publicly available and thus failed to download.

All successfully downloaded runs were trimmed using fastp v0.23.1 (Chen et al., 2018), requiring

a minimum quality score of 20 and a minimum read length of at least 25 bp (-q 20 -l 25). Trimming

results were compiled using multiqc v1.7 (Ewels et al., 2016). All trimmed runs were then aligned

to a decoy-aware transcriptome index made by combining the primary transcripts of the Araport11

genome annotation (Cheng et al., 2017) with the A. thaliana genome in salmon v1.2.1 (Patro et al.,

2017) using an index size of 25bp. The salmon outputs of each run were then combined with a

custom R script to create an gene-by-run expression matrix. We omitted 423 runs with a mapping

rate < 1 %, 215 runs with zero mapped transcripts, and 18 genes with zero mapped transcripts

across all runs from further analysis. We note that although this cut-off does not exclude samples

with more modest mapping rates (e.g. 20 - 60 %) the choice to include these samples was to avoid

removing large chunks of data as “outliers” and analyzing only those samples that conform to our

expectations.

Whole genome sequence data processing

We downloaded whole genome sequencing data for 1135 A. thaliana accessions from the 1001

genomes project panel (SRA project SRP056687) (Alonso-Blanco et al., 2016) using the SRA

toolkit. All runs were trimmed using fastp (Chen et al., 2018), requiring a minimum quality score

8

of 20 and a read length of at least 30 bp (-q 20 -l 30). Trimmed reads were then aligned to the

A. thaliana reference genome using BWA v0.7.17 (Li and Durbin, 2009). The alignments were

sorted and converted to BAM format with SAMTOOLS v1.11 (Danecek et al., 2021), then optical

duplicates were marked with picardtools v2.22.1. Haplotypes were called for each accession, then

combined and jointly genotyped with GATK v4.1.4.1 assuming a sample ploidy of 2, heterozygosity

of 0.001, indel-heterozyogsity of 0.001, and minimum base quality score of 20. Invariant sites were

included in the genotype calls with the –include-non-variant-sites option. All calls were restricted

to only coding sequence (CDS) regions based on the Araport11 annotation by supplying a BED file

of CDS coordinates made with bedtools (v2.29.2). Following Korunes and Samuk (2021), variant

and invariant sites were filtered separately using both GATK and vcftools v0.1.15 (Danecek et al.,

2011). Variant sites were filtered if they met any of the following criteria: QD < 2, QUAL < 30,

MQ < 40, FS > 60, HaplotypeScore > 13, MQRankSum < -12.5, ReadPosRankSum < -8.0, mean

depth < 10, mean depth > 75, missing genotype calls > 20%, being an indel, or having more than

2 alleles. In the end, 1,915,859 variant sites across all coding sequences were retained for further

analysis. Invariant sites were filtered if they met any of the following criteria: QUAL > 100, mean

depth < 10, mean depth > 75, missing genotype calls > 20%. Finally, variant sites were annotated

using snpEff (Java v15.0.2) (Cingolani et al., 2012b) and variants labeled as either missense or

synonymous were separated into different files using SnpSift (Cingolani et al., 2012a).

Selection estimated from between-species divergence

We identified 1:1 orthologs between the primary transcripts of A. thaliana and Arabidopsis

lyrata with Orthofinder v2.5.4 (Emms and Kelly, 2019). For each 1:1 ortholog, we aligned their

protein sequences with MAFFT L-INS-I v7.475 (Katoh and Standley, 2013), then converted the

protein alignments to gapless codon-based alignments using pal2nal v14 (Suyama et al., 2006).

Using the gapless codon-based alignments, we estimated dN /dS using the Nei and Gojobori (1986)

method implemented as a custom Biopython v1.79 script and implemented through the codeml

program in the PAML package v4.9 (Yang, 2007). Unlike codeml, the custom Biopython script also

returns counts of nonsynonymous (N ) and synonymous sites (S) within each gene as described in

9

Nei and Gojobori (1986), which we later used to calculate nucleotide diversity per nonsynonymous

site (πN ) and per synonymous site (πS). Before proceeding with more analyses, we confirmed that

our estimates of dN and dS were consistent between our Biopython script and codeml (Figure A5,

Pearson correlations dN : ρ = 0.9998, dS : ρ = 0.9809). The outputs of the Biopython script

were used in all subsequent analyses.

Selection estimated from within-species polymorphism

Nucleotide diversity (π) was calculated for each gene with pixy v1.2.3.beta1 (Korunes and

Samuk, 2021) three times: once using all sites (both variant and invariant), once using missense

sites plus invariant sites, and once using synonymous sites plus invariant sites. These estimates

were then converted to π, πN , and πS, respectively, by first multiplying the per site estimate output

from pixy by the number of sites included in the analysis. Then, to get πN and πS, the values from

analyses of missense plus invariant, and synonymous plus invariant sites were divided by the N

and S values for each gene, respectively, as determined by the method in Nei and Gojobori (1986).

We next calculated Tajima’s D for each gene. First, we calculated π and Watterson’s Theta (θW )

for each variant site i within a gene (πi and θW i respectively). In this case, πi was calculated as:
) (

)

(

πi =

ni
ni − 1

2∑

j=1

1 −

p2
ij

(1.1)

Where ni is the number of sequenced chromosomes with non-missing genotypes for variant i, pi1

is the frequency of the reference allele, and pi2 is the frequency of the alternative allele. Then, θW i

was calculated as:

Where ai is:

θW i =

1
ai

ai =

ni−1∑

j=1

1
j

(1.2)

(1.3)

This calculation of θW i is equivalent to the usual calculation of θW with the number of segregating

10

sites set to one. Next, the variance in Tajima’s D was calculated for each site as:

V ar(πi − θW i) =

− 1
ai

ni+1
3(ni−1)
ai

(1.4)

This is equivalent to equation 38 in Tajima (1989) with the number of segregating sites set to one.

Finally, the results of the above calculations were combined in the following formula:

Di =

√

πi − θW i
V ar(πi − θW i)

(1.5)

To get Tajima’s D for each gene, we then averaged across the Di values for all the variant sites

within a gene.

We also calculated the direction of selection (DoS) statistic (Stoletzki and Eyre-Walker, 2011)

for each gene. Counts of nonsynonymous and synonymous polymorphisms within each gene (PN

and PS, respectively) were determined with bedtools (v2.29.2). The number of nonsynonymous

and synonymous differences (DN and DS, respectively) between A. thaliana genes and their 1:1 A.

lyrata orthologs, if present, were estimated during the process of calculating dN /dS in Biopython

as described above. We then combined these components into the following equation:

DoS =

DN
DN + DS

− PN

PN + PS

(1.6)

We chose this metric, as opposed to the proportion of amino acid substitutions driven by positive

selection (α), because it is less biased than α (Stoletzki and Eyre-Walker, 2011) and was suc-

cessfully used in studies similar to ours (Paape et al., 2013). Furthermore, we found that α often

returns uninterpretable negative values when applied to A. thaliana, perhaps because of an excess

of slightly deleterious polymorphisms (Nordborg et al., 2005) due to their predominantly selfing

mating system (Charlesworth, 1994).

Treatment specificity

Treatment specificity (τ ) was estimated separately for runs from each tissue type using the

following formula (Yanai et al., 2005):

∑

τ =

N

i=1 1 − xi
N − 1

max x

11

(1.7)

Where x is the vector of average expression values of a gene in each treatment category, measured

in transcripts per million (TPM), and where N is the number of treatment categories. Dividing

by N means that τ varies between zero and one, where zero indicates no specificity and one in-

dicates exclusive specificity to a single treatment. We used this metric of specificity because it is

consistently more robust than others (Kryuchkova-Mostacci and Robinson-Rechavi, 2017) and is

normalized by the number of treatments included, making it comparable across data sets. We also

applied the same formula to calculate tissue specificity in several different treatment conditions.

Simulating correlations between average expression and specificity index

Average expression level and measures of expression specificity are correlated by definition

because genes with more treatment/tissue-specific expression will have lower average expression

across all treatment/tissue categories. We ran two simulations to better illustrate the factors driving

the correlation between average expression and the specificity index, τ . In both simulations, we

generated 1000 random matrices, where each element xij represented the expression of gene i in

experiment j, by sampling from a zero-inflated negative binomial distribution:

xij ∼ ZIN egBinom(N, p1, p2)

(1.8)

Where the size and probability parameters of the negative binomial component were N = 100

and p1 = 0.1, respectively, while the probability of an expression value being non-zero was p2 =

0.4. All matrices included 5 groups of columns, with 5 columns per group, representing replicates

of tissue/treatment groups. For both simulations, we averaged across columns within each group to

simulate the calculation of tissue/treatment-wide averages. We then applied the formula for τ across

the rows of this averaged matrix to get expression specificity. In one simulation, we calculated

expression level by averaging across the rows of the expression matrix. In a second simulation, we

excluded experiments where a gene was not expressed (xij = 0) from the calculation of average

expression.

12

Average expression, length, GC content, family size

Calculating the average expression of each gene was a three-step process. First, we averaged

together runs with matching SRA experiment IDs because these runs represented technical repli-

cates of the same biological sample and treatment conditions. Second, we partitioned our gene-by-

experiment expression matrix by the tissue type each sample came from. Finally, for each tissue

type’s expression matrix, we averaged across all of the expression values of each gene across all

experiments, excluding values < 5 transcripts per million (TPM). We excluded values < 5 TPM

from the average expression calculation to avoid a high correlation between average expression

and treatment-specificity, as has been reported in previous studies (Slotte et al., 2011). This high

correlation occurs because an environment-specific gene will by definition also have low average

expression across environments it is rarely expressed in. Furthermore, we excluded values < 5

TPM to avoid including small expression values that could be artifacts of alignment error.

The length and GC content of each gene was measured using the bedtools nuc command (v2.29.2)

and included each gene’s introns and untranslated regions when present. We included introns and

untranslated regions in the estimate of gene length because they play important roles in determin-

ing rates of protein evolution (Castillo-Davis et al., 2002; Eisenberg and Levanon, 2003). Finally,

the family size for each gene was estimated as the number of A. thaliana genes in their respective

orthogroups output by OrthoFinder.

Partial correlation analysis

Not all treatment-tissue combinations were sampled in the overall RNA-seq dataset, causing

confounding between the treatment and tissue labels. We resolved this in two ways. First, we

subset the data to only the treatment conditions where all tissue types were represented. Second,

we subset the data by tissue type and analyzed each subset separately. For each subset, we calculated

partial spearman correlations between treatment specificity and our measures of selection (dN , πN ,

Tajima’s D, and DoS) after accounting for average expression (excluding values TPM < 5), gene

length, and GC content using the ppcor R package (Kim, 2015). For partial correlation analyses

involving πN and Tajima’s D, we also controlled for gene family size. We did not account for gene

13

family size in partial correlation analyses involving dN or DoS because these metrics apply to

only genes with one family member in this study. When calculating partial correlations involving

dN , we excluded any genes with saturating divergence (dS > 1). All statistical analyses and

data visualizations used R v4.0.3 and used color palettes in the scico R package (Crameri, 2018;

Pedersen and Crameri, 2023).

Surrogate variable analysis

We recalculated treatment specificity and repeated all partial correlation analyses after correct-

ing each data subset for technical between-experiment variation (i.e. batch effects), following an

approach from (Fukushima and Pollock, 2020). Batch effects include variables that influence gene

expression measurements but are not of interest to this study, such as the sequencing platform and

the library prep protocol used in each experiment. First, with our data already subset by tissue type,

we further subset to only include treatments with RNA-seq runs from at least two studies. This min-

imizes confounding between-treatment variation with the technical between-experiment variation

we aimed to account for. We then applied surrogate variable analysis (SVA) using the svaseq()

function within the SVA package (Leek and Storey, 2007) to each of these subsets. Briefly, SVA

models gene expression as:

xij = µi + f (yi) + eij

(1.9)

Where xij is the expression of gene i in experiment j, µi is the average expression of gene i across

all experiments, and yi is the value of a predictor variable of interest for gene i. Furthermore, f (yi)

gives the deviation of gene i from its average expression based on the value of yi and eij is the

residual error. SVA takes this model and partitions the residual variance, eij, into:

L∑

xij = µi + f (yi) +

ℓ=1

γℓigℓj + e∗

ij

(1.10)

∑

Where

L

ℓ=1 γℓigℓj gives the summed effects of L unmodeled variables (gℓj) for each gene and e∗

ij

gives the gene-specific noise in expression. SVA does not attempt to estimate what the unmodeled

variables influencing expression are, but rather find a set of vectors (the surrogate variables) that

14

span the same space as g:

xij = µi + f (yi) +

K∑

k=1

λkihkj + e∗

ij

(1.11)

Where each hk is a surrogate variable and each λk gives the effects of each surrogate variable on

gene expression. For our analyses, our predictor variable yi was treatment type. To get a measure of

expression where the effects of surrogate variables are removed, we then subtracted off the effects

of surrogate variables from both sides of the above equation.

K∑

xij −

λkihkj = µi + f (yi) + e∗

ij

(1.12)

k=1

Where xij −

∑

K
k=1 λkihkj gives us our expression values accounting for the effects of surrogate

variables. The net result here is a reduction in the amount of unexplained or seemingly stochastic

variation in expression because sources of variation have been attributed to “surrogates” that span

the same space as real batch variables. We also conducted principal component analysis in R before

and after SVA to verify the removal of batch effects.

Results

Summary of tissue differentiation, treatment specificity, and selection in overall dataset

To understand how treatment specificity of gene expression affects evolutionary rates of pro-

teins, we queried the Sequence Read Archive for all A. thaliana RNA-seq experiments published

before May 2022. We then annotated these experiments with standardized tissue and treatment on-

tology terms, manually filtered the dataset, and then processed all RNA-seq runs with a standard-

ized pipeline. The number of sequencing experiments associated with each combination of tissue

and treatment labels is summarized in Tables S1. Overall, the most sampled tissue category was

leaf (4,642 experiments) followed by root (3,348 experiments), whole plant (2,492 experiments),

seed (1,866 experiments), shoot (1,106 experiments), then fruit and flower (266 experiments). The

four most sampled treatment categories were control (5,701 experiments), cold air exposure (675

experiments), short day length (561 experiments), and short day length plus Botrytis cinerea expo-

sure (407 experiments). Any sequencing runs that shared an SRA experiment ID were averaged to

15

produce individual gene expression values for each SRA experiment.

We first looked at the distribution of mapping rates across all RNA-seq runs. The median

mapping rate was 72.39 % (Figure A1) and we excluded runs with a mapping rate < 1% from further

analyses. We next performed a principal components analysis (PCA) on the expression matrix and

observed strong differentiation between root and non-root tissues along PC2 (Figure 1.1). We also

observed that nearly all genes had some degree of treatment specificity in their expression (Figures

1.2A, S3). Furthermore, only a small proportion of genes had strong signatures of selection based

on dN /dS, πN /πS, DoS, or Tajima’s D (Figure 1.2B-D, Figure A2). The treatment specificity of

expression was lower on average in flower and fruit tissue compared to the other tissues (Figure

A3). However, tissue specificity did not vary widely depending on the treatment condition (Figure

A4).

16

Figure 1.1: Principal components analysis of all expression data. Each point represents a dif-
ferent RNA-seq experiment and is colored by its associated tissue type. Experiments from all
treatment conditions are included in this analysis. Plotting order was randomized to avoid over-
plotting.

17

0100−100−50050PC1:  3.97 %PC2:  2.49 %Tissuefruit OR flowerleafrootseedshoot systemwhole plantFigure 1.2: Density plots of key variables measured in this study. (A) Distribution of treatment
specificity in leaf tissue expression across all genes included in this study. The area underneath the
curve in a given interval of treatment specificity represents the proportion of genes in this study that
fall within that range of treatment specificity. (B) Distribution of dN /dS across all genes included
in this study. The area to the right of the dashed line represents the proportion of genes in this study
with dN /dS > 1. (C) Distribution of πN /πS across all genes included in this study. The area to
the right of the dashed line represents the proportion of genes in this study with πN /πS > 1. (D)
Distribution of DoS across all genes in this study. Area to the right of the dashed line represents
the propotion of genes with DoS > 0, which is interpreted as evidence of adaptive evolution.

18

012340.40.60.81.0Treatment-specificity (leaf tissue)DensityA0.00.20.4-6-4-20log(dN/dS)DensityB0.00.10.20.30.4-505log(πN/πS)DensityC0123-1.0-0.50.00.5Direction of Selection (DoS)DensityDOmitting samples with low expression disentangles expression level and specificity

Genes that are only expressed in one treatment or tissue will, by definition, have low mean

expression across all environments or tissues (Wright et al., 2004). Thus, we sought a method of

calculating expression level that was independent of treatment specificity. To better understand

the relationship between average expression and treatment specificity, we calculated correlations

between treatment-specificity and expression level while either including or excluding low expres-

sion values (TPM < 5) on our real RNA-seq dataset. We found that excluding low expression values

decreased the correlation between average expression and treatment-specificity in leaf tissue sam-

ples (Figure 1.3) and other tissues (Figures A34 - A38) and replicated the result by simulating gene

expression matrices (Figure A39). Thus, for all later partial correlation analyses (see next section)

we quantified each gene’s average expression after dropping experiments where the gene was not

expressed (TPM < 5).

19

Figure 1.3: Correlation between the average expression in transcripts per million (TPM) and
treatment specificity of genes. Figures show correlations when samples with low expression (< 5
TPM) are included (A) vs excluded (B). Expression level and treatment specificity were calculated
using only data from leaf tissue samples. Line is a smoothing line with 95 % confidence intervals
and values in parentheses give spearman correlation.

Treatment specificity correlates with levels of nonsynonymous diversity and divergence in
genes

We next calculated partial correlations between treatment specificity and measures of selection

after controlling for average expression, gene length, GC content, and tissue specificity in expres-

sion. These partial correlations were calculated separately for expression data on each tissue type

and did not account for batch effects (see next section). Among leaf tissue samples, average ex-

pression had significant partial correlations with dN (ρ = −0.19, p-value = 2.1 × 10−122) and

πN (ρ = −0.17, p-value = 2.8 × 10−175) after controlling for other factors (Figures 1.4A,1.4B).

Treatment specificity was more strongly correlated with dN (ρ = 0.10, p-value = 7.6 × 10−31)

and πN (ρ = 0.10, p-value = 1.2 × 10−62) than Tajima’s D (ρ = 0.01, p-value = 3.1 × 10−7)

and DoS (ρ = 0.04, p-value = 2.3 × 10−06, Figure 1.4C, 1.4D). Furthermore, the top 25% most

20

−10−505100.40.60.81.0treatment specificity of expressionlog(average expression in TPM)leaf (−0.63)A2.55.07.510.00.40.60.81.0treatment specificity of expressionlog(average expression in TPM)leaf (0.16)Btreatment-specific genes in leaf tissue for our dataset have average dN and πN values nearly 2.5

times greater than the 25% least treatment-specific genes (dN = 0.025 vs 0.061; πN = 0.0014 vs

0.0032). Meanwhile, the most and least treatment-specific genes have average Tajima’s D values

of are -0.44 and -0.43, respectively, and average DoS values of -0.19 and -0.14, respectively. The

strongest partial correlation generally occurred between tissue specificity and treatment specificity

(Spearman’s ρ = 0.53 − 0.60, Figure 1.4). Gene family size had among the weakest partial cor-

relations with πN compared to other covariates, but strongly correlated with treatment specificity

(ρ = 0.12, p-value = 6.3 × 10−84, Figure 1.4B). All of these findings generally held when average

expression and treatment specificity were calculated on data from other tissues (Table S2, Figures

A6-A10).

21

Figure 1.4: Partial correlation analysis between specificity and selection. Analyses include
either (A) dN , (B) πN , (C) Tajima’s D, or (D) direction of selection (DoS) as a covariate. Average
expression excludes values < 5 TPM and was calculated using only leaf tissue samples. Treatment
specificity was also calculated using only leaf tissue samples. Tissue specificity was calculated
using only control samples across all tissue categories. The number of genes included in each
partial correlation analysis (n) is listed at the top of each heatmap.

22

-0.40.00.4Partialspearmancorrelationcoefficient0.170.11-0.020.10.050.53-0.190.02-0.10.09-0.030.08-0.12-0.01-0.15-0.18-0.14-0.15-0.15-0.35-0.12dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 14629A0.540.14-0.080.1-0.020.58-0.170.1-0.150.11-0.02-0.02-0.10.01-0.13-0.130.08-0.16-0.21-0.34-0.160.020.060.050.120.070.030.11πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 26497B0.010.030.6-0.01-0.180.09-0.02-0.10.01-0.130.01-0.18-0.23-0.33-0.160.050.050.120.070.030.11Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 26503C0.050.040.55-0.04-0.130.07-0.05-0.12-0.01-0.15-0.06-0.18-0.18-0.33-0.14Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 14573DCorrelations between treatment specificity and nonsynonymous variation persist after
controlling for batch effects and dataset imbalance

While combining gene expression data across multiple studies can increase the statistical power

of an analysis, there are some potential concerns. First, if many tissue-treatment combinations are

not sampled, the dataset will be unbalanced and the effects of tissue and treatment variation on

expression could be confounded. Consistent with this expectation, there was a high correlation

between tissue specificity and treatment specificity in our initial analyses (Figure 1.4, S6-S10).

Furthermore, combining data from multiple laboratories could generate batch effects (Leek et al.,

2010). To address the issues of imbalance and batch effects, we first subset our data to only include

treatments where all tissue types were represented. This subset included the treatments of control,

abscisic acid, continuous light, warm/hot air temperature, and cold air temperature. We then used

SVA to correct for the influence of unknown batch effects on this data subset (Leek and Storey,

2007). After SVA, treatment specificity positively correlated with dN (ρ = 0.10, p-value = 1.6 ×

10−32) and πN (ρ = 0.07, p-value = 1.5×10−23) when average expression and treatment specificity

were calculated on combined fruit and flower data (Figures A33). However, treatment specificity

in other tissue types generally did not correlate with our measures of selection (Figures A28-S33,

Table S4).

The inclusion of only five treatments in the above analysis could limit quantification of a gene’s

treatment specificity. Thus, in order to include data from a larger number of treatments, avoid

dataset imbalance, and avoid batch effects, we split our expression matrix into six subsets by tissue

category. We then further removed treatments that only had expression data from one study to avoid

confounding treatment effects with study-specific batch effects. We applied SVA (Leek and Storey,

2007) to each of these tissue-specific subsets. After SVA, the expression profiles of most genes

appear less treatment-specific (Figures A16-A21 panels A vs B). We also observed less separation

in PCA space within treatment groups after SVA (for example, see Figures A16C and A16D).

Average expression levels before SVA were generally correlated with expression levels after SVA

(Figures A16-A21 panels A and B). In partial correlations on each SVA-corrected subset, treatment

specificity significantly correlated with dN (ρ = 0.13, p-value = 6.9 × 10−50) and πN (ρ = 0.16, p-

23

value = 3.9 × 10−128) but less strongly correlated with Tajima’s D (ρ = 0.04, p-value = 6.6 × 10−10)

and DoS (ρ = 0.05, p-value = 2.0 × 10−8) for the leaf tissue data subset (Table 1, Figures 1.5).

These patterns were similar in other tissue types (Figures A11-A15, Table S3).

Pre/post-SVA

Selection measure

Pre
Post
Pre
Post
Pre
Post
Pre
Post

dN
dN
πN
πN
Tajima’s D
Tajima’s D
DoS
DoS

Partial correlation:
selection vs
treatment-
specificity
0.10
0.13
0.10
0.16
0.03
0.04
0.04
0.05

p-value

7.6 × 10−31
6.9 × 10−50
1.2 × 10−62
3.9 × 10−128
3.1 × 10−7
6.6 × 10−10
2.3 × 10−6
2.0 × 10−8

Table 1: Partial correlations between treatment-specificity and different measures of selection
pre-SVA and post-SVA. All correlation coefficients are spearman coefficients and are calculated
only on leaf tissue samples. All p-values represent whether correlation coefficient significantly
differs from 0.

24

Figure 1.5: Partial correlations between specificity and selection after SVA. Analyses include
(A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on leaf tissue data
subset after applying SVA. Data was further subset to include only treatment groups with data from
more than one study before applying SVA. Average expression calculation excludes values < 5
TPM. The number of genes included in each partial correlation analysis (n) is listed at the top of
each heatmap.

25

-0.40.00.4Partialspearmancorrelationcoefficient0.160.130.03-0.270.02-0.05-0.050.08-0.1-0.14-0.2-0.17-0.25-0.28-0.08dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 13133A0.530.16-0.06-0.260.15-0.05-0.04-0.01-0.07-0.12-0.150.09-0.34-0.24-0.1-0.010.070.10.040.010.07πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 22459B0.040-0.09-0.03-0.08-0.120.01-0.37-0.22-0.10.050.10.040.010.08Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 22461C0.05-0.07-0.09-0.05-0.1-0.14-0.06-0.3-0.25-0.09Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 13087DDiscussion

Our main finding is that genes with more treatment-specific expression patterns are, on aver-

age, under weaker selective constraint in A. thaliana. This is evident by treatment-specific genes

generally having higher values of πN and dN , but not higher values of Tajima’s D and DoS, com-

pared to genes with more constitutive expression (Figures 1.4,1.5). Our result does not refute the

possibility of strong positive selection on treatment-specific genes, as is the case for nucleotide

binding site leucine rich repeat proteins (NBS-LRRs) in A. thaliana (Mondragón-Palomino et al.,

2002). Rather, treatment-specific genes are simply under weaker selection on average compared to

less treatment-specific genes. Altogether, this pattern is consistent with the hypothesis that a trade-

off between the strength of selection and the treatment specificity of expression helps maintain

variation in plasticity for A. thaliana (Snell-Rood et al., 2010; Van Dyken and Wade, 2010).

There are a few ways to think about the biological relevance of the correlations of treatment

specificity with πN and dN . First, the magnitude of treatment specificity’s correlation with πN

and dN was generally half the magnitude of average expression’s correlation with πN and dN and

similar to tissue specificity’s correlation with πN and dN . Both tissue specificity and average ex-

pression are thought to be important determinants of protein evolution (Bush et al., 2015; Wu et al.,

2022), suggesting the comparable effects of treatment specificity may be important too. Second,

the effect of treatment specificity on πN and dN persisted even after simultaneously controlling

for expression level, tissue specificity, gene length, GC content, and batch effects. Finally, the top

25% most treatment-specific genes in our dataset have average dN and πN values nearly 2.5 times

greater than the 25% least treatment-specific genes (dN = 0.025 vs 0.061; πN = 0.0014 vs 0.0032),

but relatively similar Tajima’s D and DoS values (Tajima’s D = -0.44 vs -0.43; DoS = -0.19 vs

-0.14). These observations together suggest that treatment specificity is an important determinant

of protein evolution.

This study disentangles several processes that were often difficult to resolve in previous re-

search. First, many previous studies focus mainly on explaining trends in dN /dS (Slotte et al.,

2011; Gaut et al., 2011; Bush et al., 2015), but both relaxed negative selection and increased posi-

26

tive selection can lead to increases in dN /dS. To tease apart these two processes, we additionally

investigated treatment specificity’s relationship with Tajima’s D and DoS. Treatment specificity’s

weaker correlation with Tajima’s D and DoS, compared to dN and πN , suggests that relaxed nega-

tive selection plays a larger role than increased positive selection in explaining the high evolution-

ary rates of treatment-specific genes. Furthermore, measures of expression specificity are often

highly correlated with expression level (Slotte et al., 2011; Alvarez-Ponce and Fares, 2012; Huang,

2022). When calculating a gene’s expression level, we only included samples where said gene was

expressed (TPM > 5) to get an estimate of expression level that was still correlated with dN and

πN , but was independent of expression specificity, allowing us to better disentangle these factors.

Finally, previous studies have struggled to partition the factors that influence selection on genes in

the presence of predictor variables with considerable error, such as expression level (Drummond

et al., 2006; Plotkin and Fraser, 2007; Yang and Gaut, 2011). Error in expression measurements can

often be attributed to unmeasured differences between RNA-sequencing experiments (Leek et al.,

2010) and we accounted for these differences using SVA (Leek and Storey, 2007). Even after SVA,

treatment specificity was strongly correlated with dN and πN (Figures 1.5A-B), suggesting our re-

sults are not an artifact of errors in expression measurement or combining expression data across

many studies.

Surprisingly, nearly all genes in A. thaliana have some degree of treatment specificity in their

expression (Figures 1.2A, A3), reflecting results of previous studies on tissue specificity (Eisen-

berg and Levanon, 2003). The high prevalence of treatment specificity in our dataset is partly

explained by batch effects because SVA significantly lowered the apparent treatment specificity of

most genes (Figures A16B-A21B) and reduced within-treatment differentiation in PCA space (for

example, see Figures A16C and A16D). This reduction in treatment-specificity likely happened

because batch effects can include unrecorded between-treatment differences (e.g.

the humidity

of the growth chamber, light intensity, watering schedule, etc.). Controlling for these unrecorded

between-treatment differences thus causes the expression of genes to be less treatment-specific.

However, even after batch correction most genes still showed some degree of treatment specificity

27

(Figures A16B-A21B), suggesting it is rare for a gene to be expressed at the same level across many

environments.

We also observed that genes with higher treatment specificity generally belonged to larger gene

families. We expected gene family size to correlate with selection because singleton and duplicated

genes often evolve at different rates (Jordan et al., 2004; Davis and Petrov, 2004). Theory also

suggests that gene duplication leads to relaxation of selection on duplicates, allowing for neo- and

sub-functionalization (Lynch and Conery, 2000; Aagaard et al., 2006). We could not investigate

how gene family size correlates with dN or DoS because measuring these quantities requires iden-

tifying substitutions between orthologous genes. Thus, dN and DoS can only be reliably measured

for 1:1 orthologs between A. thaliana and A. lyrata. However, πN and Tajima’s D can be calcu-

lated for genes in larger families and we did observe persistent correlations between family size

and Tajima’s D (For Figure 1.5C: ρ = 0.05, p-value = 3.1 × 10−12; also see Figures A6C-A15C,

A28C-A33C). Altogether, these correlations suggest that processes of gene duplication, neofunc-

tionalization, and subfunctionalization could be connected to evolving some degree of treatment

specificity.

Gene length was generally the second most correlated factor with dN and πN in our study, just

behind average expression. This is consistent with previous work suggesting that longer proteins

require more energy to synthesize and are thus under stronger selective constraints (Urrutia and

Hurst, 2001; Castillo-Davis et al., 2002; Eisenberg and Levanon, 2003; Urrutia and Hurst, 2003).

However, while some previous studies in A. thaliana observe this same trend (Bush et al., 2015),

others do not (Slotte et al., 2011). This discrepancy could be due to differences in how gene length

is defined between studies. In this study, each gene’s length included coding sequence as well

as introns and untranslated regions, whereas other studies break down gene length into individual

features (Bush et al., 2015). The goal of this study was not to understand differences in evolution

between different gene features, so we included all gene features in our estimate of gene length.

However, introns and untranslated regions experience different evolutionary patterns than coding

sequences; for example, highly expressed genes being under selection for shorter introns (Castillo-

28

Davis et al., 2002; Eisenberg and Levanon, 2003). Therefore, future studies must clearly define

even seemingly simple features like gene length to ensure that results are comparable across studies.

Although we focused on testing the idea that treatment specificity is responsible for relaxed

negative selection in some genes, it is also possible that relaxed selection caused the evolution of

treatment specificity. There is some evidence that relaxation of selection occurs before the evolu-

tion of expression specificity (Hunt et al., 2011) and may better explain cases of neo- and subfunc-

tionalization (Lynch and Conery, 2000; Aagaard et al., 2006). Future experiments that look at the

evolution of treatment specificity and sequence evolution across a broader phylogenetic scale may

be helpful for determining the order of these processes.

In summary, this study investigates a trade-off between the treatment-specific expression of a

gene and the strength of selection said gene experiences, which is hypothesized to limit plasticity

evolution. Consistent with this hypothesis, genes in A. thaliana with more treatment-specific ex-

pression are under weaker selection compared to more evenly expressed genes. While we find that

this trade-off exists, we could not dissect the direction of causality in the trade-off or determine how

much this trade-off constrains plasticity evolution relative to other processes. However, these are

exciting areas of future research. Future studies should ideally generate fully balanced datasets on

gene expression acquired across natural environmental gradients. Taking these steps will contribute

to a comprehensive understanding of the constraints on plasticity and protein evolution.

Data availability

Supplemental figures are available in Appendix A. All code for our bioinformatic workflows,

data analysis, and figure creation can be found here: https://github.com/milesroberts-123/arabidopsis-

conditional-expression. All supplementary tables are available online at: https://doi.org/10.1093/

genetics/iyad074. The tissue type and treatment annotations for RNA-seq runs in our study can

be found in Table S5. Genomic references, mapping rates, and a table of expression specificity;

nucleotide diversity; and substitution rate values estimated for all A. thaliana genes included in

this manuscript’s analyses is available at: https://doi.org/10.5061/dryad.xd2547dnd. The genome

assembly and annotation used in this study was originally downloaded from Phytozome: https:

29

//phytozome-next.jgi.doe.gov/.

30

CHAPTER 2: k-MER-BASED DIVERSITY SCALES WITH POPULATION SIZE
PROXIES MORE THAN NUCLEOTIDE DIVERSITY IN A META-ANALYSIS OF 98
PLANT SPECIES2

Abstract

A key prediction of neutral theory is that the level of genetic diversity in a population should

scale with population size. However, as was noted by Richard Lewontin in 1974 and reaffirmed

by later studies, the slope of the population size-diversity relationship in nature is much weaker

than expected under neutral theory. We hypothesize that one contributor to this paradox is that

current methods relying on single nucleotide polymorphisms (SNPs) called from aligning short

reads to a reference genome underestimate levels of genetic diversity in many species. As a first

step to testing this idea, we calculated nucleotide diversity (π) and k-mer-based metrics of genetic

diversity across 112 plant species, amounting to over 205 terabases of DNA sequencing data from

27,488 individuals. After excluding 14 species with low coverage or no variant sites called, we

compared how different diversity metrics correlated with proxies of population size that account

for both range size and population density variation across species. We found that our popula-

tion size proxies scaled anywhere from about 3 to over 20 times faster with k-mer diversity than

nucleotide diversity after adjusting for evolutionary history, mating system, life cycle habit, cul-

tivation status, and invasiveness. The relationship between k-mer diversity and population size

proxies also remains significant after correcting for genome size, whereas the analogous relation-

ship for nucleotide diversity does not. These results are consistent with the possibility that variation

not captured by common SNP-based analyses explains part of Lewontin’s paradox in plants, but

larger scale pangenomic studies are needed to definitively address this question.

Introduction

Understanding the determinants of genetic diversity within populations is key to informing

species conservation (Cole, 2003) and breeding efforts (Sanchez et al., 2023). However, most

species have far less genetic diversity (commonly estimated as pairwise nucleotide diversity, π)

than expected (Frankham, 2012; Corbett-Detig et al., 2015; Buffalo, 2021).

If we assume that

2This chapter is, as of writing, accepted for publication at Evolution Letters. The preprint version is currently

available at the following DOI under a CC-BY License: https://doi.org/10.1101/2024.05.17.594778

31

the vast majority of genetic variants are neutral, then the determinants of genetic diversity are en-

capsulated in neutral theory (Kimura, 1983): E[π] ≈ 4Neµ, where E[π] is the expected level of

genetic diversity, Ne is the effective size of a population, and µ is the mutation rate per base pair per

generation. Mutations rates for SNPs and small indels vary relatively little across species (Cagan

et al., 2022; Bergeron et al., 2023; Quiroz et al., 2023), while the total number of individuals in

a species varies massively (Buffalo, 2021). Thus, under neutral theory, population size should be

a strong determinant of genetic diversity and species with larger population sizes should be more

diverse. However, even some of the most abundant species studied to date have low genetic diver-

sity compared to neutral theory expectations. For example, Drosophila simulans has an estimated

population size > 1014 and a diversity of π ≈ 0.01, but an expected diversity of π > 0.1 (Buffalo,

2021). This mismatch between expected and observed levels of neutral diversity across populations

of varying size is known as Lewontin’s paradox, named after Richard Lewontin who first described

the phenomenon (Lewontin, 1974).

The potential mechanisms underlying Lewontin’s paradox have been reviewed extensively

(Leffler et al., 2012; Slotte, 2014; Ellegren and Galtier, 2016; Charlesworth and Jensen, 2022). Mul-

tiple selective and demographic processes likely contribute to Lewontin’s paradox; however, deter-

mining the relative importance of these processes remains a contentious area of research. The two

most explored mechanisms are historic population size changes (i.e. demography, Charlesworth

and Jensen (2022)) and linked selection - whereby fixation or purging of selected alleles causes the

loss of linked neutral alleles (Kojima and Schaffer, 1964, 1967; Smith and Haigh, 1974; Charlesworth

et al., 1993; Charlesworth, 1994). Linked selection is expected to reduce diversity more in regions

of lower recombination and higher functional density (Slotte, 2014) and many studies have tested

this hypothesis(Tenaillon et al., 2001; Hellmann et al., 2003; Nordborg et al., 2005; Roselius et al.,

2005; Branca et al., 2011; Paape et al., 2012; Corbett-Detig et al., 2015; Silva-Junior and Grat-

tapaglia, 2015; Wang et al., 2016; Phung et al., 2016; Mackintosh et al., 2019). These previous

investigations often conclude that linked selection contributes to Lewontin’s paradox, but not all

report significant results (e.g. Schmid et al. (2005); Roselius et al. (2005); Flowers et al. (2012);

32

Wang et al. (2016)). It’s been argued that studies focused on plant species especially tend to find

weaker evidence for linked selection (Slotte, 2014). There is also both empirical and theoretical

evidence that linked selection is unlikely to explain the entirety of Lewontin’s paradox, suggesting

that demographic factors play an important role too (Coop, 2016; Buffalo, 2021; Charlesworth and

Jensen, 2022).

There are three main types of demographic changes proposed to contribute to Lewontin’s para-

dox: contractions, expansions, and cyclical population size changes (Charlesworth and Jensen,

2022). Population contractions cause loss of diversity. Thus, if many species’ populations recently

contracted (due to human activity, for example), then their contemporary diversity would be much

lower than expected from their pre-contraction population sizes (Exposito-Alonso et al., 2022).

Recent population expansions could cause a similar mismatch. Because it takes many generations

for populations to accumulate diversity compared to the timescale of typical expansions, contem-

porary diversity levels for an expanded population would be much smaller than expected from a

post-expansion population size (Peart et al., 2020; Charlesworth and Jensen, 2022). For a similar

reason, species that have seasonal variation in their population sizes will also tend to have diversity

levels closer to what one would expect based on their minimum size rather than their peak size

(Wright, 1940). Studies investigating Lewontin’s paradox would ideally try to jointly infer these

demographic histories alongside selective factors in natural populations. However, issues of model

complexity and identifiability often prevent such joint estimation (Johri et al., 2020, 2022b,a), sug-

gesting further explorations of Lewontin’s paradox will require new approaches.

Two potential, but rarely explored, contributors to Lewontin’s paradox are that (1) current meth-

ods for estimating genetic diversity systematically underestimate the true levels of genetic diver-

sity in most populations and (2) changes or corrections to the calculation of diversity could yield

stronger correlations with population size. Lewontin’s original observations and earlier studies on

the population size-diversity relationship were based on allozymes, which detect variants in pro-

tein sequences (Lewontin, 1974; Nei and Graur, 1984). More recent studies measure diversity using

SNPs at more neutral four-fold degenerate sites (i.e. sites where mutations do not affect protein

33

sequences) in DNA and generally observe greater within-species diversity and between-species

divergence compared to allozymes (Li and Sadler, 1991; Makałowski and Boguski, 1998; Bazin

et al., 2006; Piganeau and Eyre-Walker, 2009). However, current SNP-based methods are not per-

fect either and there is significant evidence that SNPs capture a biased and incomplete picture of

genetic diversity. Calling SNPs typically requires aligning reads to a reference genome, meaning

any SNPs in regions that are not present or highly diverged from the reference genome will be

excluded from analysis and thus downwardly bias diversity estimates (Golicz et al., 2020; Buffalo,

2021). This downward bias is typically assumed to have little effect on the qualitative relationship

between diversity and Ne (Buffalo, 2021), but recent pangenomic studies have uncovered troves

of non-reference variation across a variety of species (Ebler et al. (2022); Rice et al. (2023), re-

viewed in Bayer et al. (2020)). Sometimes alignment errors in regions of non-reference structural

variation will also create false positive SNP calls, which will also affect diversity estimates (e.g.

Jaegle et al. (2023)). Finally, previous meta-analyses of population size and diversity data rely on

scraping diversity estimates from previously published studies (Frankham (2012); Buffalo (2021),

except see Corbett-Detig et al. (2015)). However, many studies report inaccurate SNP calls and

deflated diversity estimates due to errors in the handling of missing genotype calls (Korunes and

Samuk, 2021; Schmidt et al., 2021; Sopniewski and Catullo, 2024) and may filter genotype calls

differently, making comparisons across species difficult. Overall, errors in diversity calculations

and omission of diversity in genomic regions that are either difficult or impossible to align to could

partially explain Lewontin’s paradox. Re-analyzing whole genome sequencing data with a common

pipeline and applying correct calculations of nucleotide diversity would make diversity estimates

across species more comparable and easier to interpret (Buffalo, 2021; Mirchandani et al., 2024).

One useful pangenomics tool for measuring non-reference variation that is readily applicable

to common short-read datasets is the k-mer. k-mers are subsequences of length k derived from a

larger sequence and they have a long history of use in computer science (Shannon, 1948), genome

assembly (Turner et al., 2018), metagenomics (Benoit et al., 2016), and quantitative genetics (Rah-

man et al., 2018; Voichek and Weigel, 2020; Kim et al., 2020; Mehrab et al., 2021). Recent studies

34

have also demonstrated the utility of k-mers for measuring heterozygosity and genetic differences

between individuals (commonly referred to as “dissimilarity” measures, Ondov et al. (2016); Vur-

ture et al. (2017); Ranallo-Benavidez et al. (2020); VanWallendael and Alvarez (2022); Roberts

et al. (2024)). Typical analysis of k-mers involves only counting the presence/absence and/or fre-

quencies of all k-mers in a set of reads, without aligning the reads to any reference, then deriving

measures of genetic difference from such counts (Benoit et al., 2016). Avoiding alignment allows

one to incorporate sequences that would otherwise be omitted for lack of alignment to a reference

genome (Rahman et al., 2018; Voichek and Weigel, 2020; Wiersma et al., 2024).

We revisited Lewontin’s paradox in plants using k-mer-based measures of genetic difference

and corrected π calculations, aiming to test whether the inclusion of non-reference variation or

modifications to diversity calculations could increase the scaling between population size proxies

and diversity. We compared how k-mer dissimilarity and typical SNP-based estimates of nucleotide

diversity correlated with population size proxies across a large panel of plant species - all processed

through the same bioinformatic pipeline. Our expectation was that if k-mers are better at capturing

genomic variation than SNPs, k-mer dissimilarity would scale more rapidly with population size

compared to nucleotide diversity.

Methods

Our entire analysis is packaged as a snakemake workflow stored here: https://github.com/milesroberts-

123/tajimasDacrossSpecies. This workflow includes the code to reproduce all of the steps individ-

ually explained below, along with instructions on how to run the code, and yaml files describing

the exact configurations of software we used at each step. It also includes an example directed

acyclic graph showing the order of steps a typical sample is processed through. The code detailing

all initial, exploratory, and confirmatory data analyses as well as figure creation can be found as

an R-markdown file in the github repository. The parameters for each software were kept constant

across all datasets (except occasionally for the “–ploidy” parameter in GATK HaplotypeCaller) to

ensure that variation in bioinformatic processing did not bias our results. All statistical analyses

used R v4.2.2 (R Core Team, 2022) and all color palettes used in figure creation come from the

35

scico R package (Pedersen and Crameri, 2023) to ensure color-blind accessibility.

Population-level sequencing data collection

We started by building a list of species with high quality, publicly available reference genomes

as well as population-level sequencing data. The source for the genome assembly and annotation

used for each species in this study is listed in Table S1. We first downloaded all genomes in Phy-

tozome (https://phytozome-next.jgi.doe.gov/) with unrestricted data usage. We then downloaded

all genomes for species from Ensembl plants (https://plants.ensembl.org/index.html) that were not

already represented in Phytozome. Next, we downloaded genomes for additional species from the

NCBI genome database (https://www.ncbi.nlm.nih.gov/genome/) that were not already present in

either Phytozome or Ensembl and met all of the following criteria:

• matched filters: eukaryotic, plants, land plants, and exclude partial

• included assemblies of nuclear DNA (i.e. not just plastid genomes)

• included annotations of coding sequences

We also downloaded a genome for Nicotiana tabaccum from the Sol genomics network (https:

//solgenomics.net/). Finally, we omitted 8 species (Aegilops tauschii, Hordeum vulgare, Lens culi-

naris, Pisum sativum, Thinopyrum intermedium, Trifolium pratense, Triticum aestivum, Triticum

turgidum) that had at least one chromosome longer than 229 bp (about 537 Mb) from all down-

stream analyses because tabix indexing, which is often utilized for SNP-calling pipelines, does not

support chromosomes exceeding this length. In the end, we were left with genome assemblies and

annotations for 112 plant species (see Table S1).

Note that, similar to previous studies (Corbett-Detig et al., 2015; Buffalo, 2021), many of the

plant species in this set of 112 are domesticated (see Table S1). This means that many of the species

in our dataset have likely undergone recent demographic changes. However correctly accounting

for demography is a general limitation of Lewontin’s paradox studies, as we can only estimate

proxies of Ne for contemporary populations (Corbett-Detig et al., 2015; Buffalo, 2021). We include

cultivation status in our downstream modeling to help account for systematic differences between

cultivated and wild species (see Statistical analysis) and also repeat our analysis with multiple

36

types of population size proxies.

For each species with a reference genome, we searched for DNA-seq runs in the National Center

for Biotechnology Information’s Sequence Read Archive (SRA) with a name in the organism field

that matched the species name (e.g. search for Arabidopsis lyrata[Organism] to get Arabidopsis

lyrata runs). We downloaded the run info for each search and found the study with most sequenced

individuals for inclusion in our analysis. Most datasets came from individual studies, with the

exception of Zea mays, which included several studies described in Bukowski et al. (2017). The

datasets used for each species are listed in Table S1.

We limited the size of each species’ dataset to no more than 7.5 × 1012 bp and no more than

1200 individuals because this defined the amount of data our workflow could process without the

peak memory limit exceeding 50 TB and the time limit for genotype calling exceeding 7 days. If

a species’ dataset exceeded either 1200 individuals or 7.5 × 1012 bp, we randomly dropped one

individual at a time until both of these limits were satisfied.

We downloaded the SRA runs associated with each individual using the SRA toolkit (v2.10.7),

then trimmed low-quality base calls with fastp (v0.23.1, Chen et al. (2018)), requiring a minimum

quality score of 20 and a minimum read length of 30 base pairs. For each species, we summarized

the results of fastp trimming using multiqc (v1.18, Ewels et al. (2016)). After trimming, any fastq

files that were technical replicates of the same individual were concatenated. Concatenated fastq

files were then processed through two different workflows: SNP-calling and k-mer counting.

SNP calling pipeline

We aligned sequencing reads for each individual to their respective reference genome using

BWA MEM (v0.7.17, Li and Durbin (2009); Li (2013)), sorted the resulting BAM files with sam-

tools (v1.11, Danecek et al. (2021)), and marked optical duplicates with picardtools (picard-slim

v2.22.1, Institute (2019)). Next, we called SNPs with GATK HaplotypeCaller (v4.1.4.1, McKenna

et al. (2010); Poplin et al. (2018)). We varied the –ploidy parameter for HaplotypeCaller be-

tween species depending on the actual ploidy recorded in the literature and whether individual

subgenome assemblies were available. However, the vast majority of species in our dataset had

37

a –ploidy paramter of 2. We restricted genotype calling to only 4-fold degenerate sites within the

nuclear genome and included all nuclear genome scaffolds, as identified by degenotate (v1.1.3,

Mirchandani et al. (2024)), to focus solely on neutral diversity. Notably, we chose to not call

SNPs in non-coding sequences in plants because plant genomes tend to be highly repetitive and

are largely composed of transposable elements, making short read mapping difficult For example,

even in the small-genome plant species Capsella grandiflora, SNPs can only be confidently called

in 10% of intergenic sites (Williamson et al., 2014). Runs for each species were then combined

with GATK GenomicsDBImport, then genotyped with GATK GenotypeGVCFs, including invari-

ant sites as done in Korunes and Samuk (2021). Variant and invariant sites were separated with

bcftools (v1.17, Danecek et al. (2021)) and then filtered separately, as recommended by Korunes

and Samuk (2021). Variant sites were removed from our analyses if they met at least one of the

following criteria: number of alleles > 2, indel status = TRUE, fraction of missing genotypes >

0.2, QD < 2.0, QUAL < 30.0, MQ < 40.00, FS > 60.0, HaplotypeScore > 13.0, MQRankSum <

-12.5, and ReadPosRankSum < -8.0. In short, these filters reflect the standard population genetics

practices of removing non-biallelic SNPs and SNPs with a lot of missing genotype calls. The exact

filter cutoffs also come from GATK Best Practices for hard filtering (Caetano-Anolles, 2023) that

is widely used (e.g. (Yu et al., 2024; Ramirez-Ramirez et al., 2024; Glasenapp and Pogson, 2024;

Ritter et al., 2024)). For each species, we also required that each variant site have a minimum read

depth of 5, but no more than 3 times the genome-wide average read depth at variant sites for that

species. We chose a minimum read depth of 5 to maintain comparability with our k-mer dissimi-

larity calculations which required minimum k-mer counts of 5 (see k-mer counting), but we also

filtered out calls with very high read depths because these could represent read mismapping due

to repetitive or paralogous regions. Meanwhile, invariant sites were removed from our analyses if

they met at least one of the following criteria: QUAL > 100.0, read depth ≤ 5, or read depth ≥ 3

times the genome-wide average read depth at invariant sites for that species. Finally, invariant and

variant sites were concatenated into a single VCF file per scaffold using bcftools. For Brassica na-

pus and Miscanthus sinensis, scaffolds named “LK032656” (195,249 bp, 0.023 % of the genome)

38

and “scaffold04645” (2,838 bp, 0.000136 % of the genome), respectively, were omitted from our

analyses because an error in SLURM job cancellation caused snakemake to prematurely delete

intermediate files for these scaffolds. It is worth noting that different choices of genotype callers

and filtering parameters could lead to different estimates of nucleotide diversity. However, our

workflow is representative of SNP calling workflows used in many published population genetic

analyses.

Using the SNP genotypes called from our pipeline, we then calculated genome-wide average

nucleotide diversity at four-fold degenerate sites (¯π) using the filtered set of variant and invariant

sites. To do this, we first calculated heterozygosity at each four-fold degenerate site (i) according

to Hahn (2018):

(

ni
ni − 1

πi =

) (

)

1 −

ai∑

j=1

p2
ij

(2.1)

where ni is the number of sequenced chromosomes with non-missing genotypes for site i, ai is the

number of alleles for site i, and pij is the frequency of the jth allele at site i. For each invariant

site, the equation reduces to πi = 0 because pi1 = 1 and ai = 1. To get ¯π, we then calculated the

average value of πi across all M sites in the genome (including both variant and invariant sites):

∑

M
i=1 πi
M

¯π =

(2.2)

Importantly, this calculation of nucleotide diversity does not assume that all missing genotype

calls are homozygous for the reference genotype, meaning that our estimates of π will not be down-

wardly biased by missing data (Korunes and Samuk, 2021).

k-mer counting pipeline

For each species, we counted 30-mers in the coding sequences of their respective reference

genome using KMC (v3.2.1, Kokot et al. (2017)). We chose to count k-mers of 30 base pairs (i.e.

30-mers) for all species in our dataset because previous k-mer-based analyses in plants typically

analyzed k-mers in the range of 20 - 40 base pairs (Voichek and Weigel, 2020; Kim et al., 2020;

39

VanWallendael and Alvarez, 2022; Ruperao et al., 2023) and because k-mers in this range can

be reliably sequenced with short reads while capturing the majority of unique genomic sequences

(Shajii et al., 2016; Ondov et al., 2016; Roberts et al., 2024). Next, we removed any 30-mers that

matched 30-mers found in each species’ corresponding set of coding sequences. This step intended

to focus our k-mers down to a set that is evolving more neutrally on average, analogously to how

we focused on only 4-fold degenerate SNPs in our SNP-calling pipeline. The justification for this

approach is that non-coding sequences generally have weaker signals of interspecies conservation

compared to coding sequences (Woolfe et al., 2005; Siepel et al., 2005; Johnsson et al., 2014).

Although, similarly to 4-fold degenerate sites, many studies have observed non-coding sequences

under selective constraints (Margulies et al., 2003; Guo et al., 2007). Thus, similar to the com-

mon analysis of 4-fold degenerate sites, our k-mer analysis is limited by an inability to completely

remove the effects of selection on sequence diversity.

Although comparing our k-mer and nucleotide diversity metrics will be affected by differences

between coding and non-coding sequences, many previous studies found that non-coding regions

and 4-fold degenerate sites have very similar levels of diversity (Moriyama and Powell, 1996;

Makałowski and Boguski, 1998; Halushka et al., 1999; Zwick et al., 2000; Tenaillon et al., 2001;

Nordborg et al., 2005; Branca et al., 2011; Williamson et al., 2014; Wang et al., 2016; Phung et al.,

2016; Mattila et al., 2017). Previous investigations of Lewontin’s paradox also found that diversity

levels across species vary much more than diversity levels across different categories of putatively

neutral sequences (Leffler et al., 2012; Buffalo, 2021) and subsequently pooled estimates of neutral

diversity across different categories of sites. Our analysis choices here are thus in line with previous

studies; however, larger scale pangenomic analyses will be helpful in relaxing this assumption.

For most species in this study, we identified hundreds of millions of unique 30-mers. It would

be computationally expensive to analyze all the 30-mers for every species. However, previous

studies have shown that one can randomly downsample k-mer sets with very minimal effects on

measures of genomic dissimilarity (Fofanov et al., 2004; Benoit et al., 2020; Roberts et al., 2024).

Thus, we randomly downsampled each species’ 30-mer list to 10 million 30-mers with a frequency

40

≥ 5 in at least one sample in the species’ 30-mer list. We chose to downsample to 10 million 30-

mers to decrease disk space burden of storing k-mer counts and because several previous studies

show that subsets of only 1 million k-mers or less can reliably estimate genetic dissimilarity in

many systems (Ondov et al., 2016; Benoit et al., 2020; VanWallendael and Alvarez, 2022). The

reason we also included a frequency cut-off of 5 is to omit k-mers containing sequencing errors,

which predominantly occur as very low-frequency k-mers (Ranallo-Benavidez et al., 2020). We

chose a frequency cutoff of 5 to to include more k-mers in some of our lower coverage datasets, but

frequency cut-offs anywhere from 2-10 are used in plants (Voichek and Weigel, 2020; VanWallen-

dael and Alvarez, 2022). We then joined the subset k-mer counts for each individual into a single

matrix for each species. We used this k-mer frequency matrix to measure genetic distance in two

ways. First, we calculated Jaccard dissimilarity (JD, Ondov et al. (2016)) between each pair of

individuals in a species’ dataset as:

JD(X, Y ) = 1 − X ∩ Y
X ∪ Y

(2.3)

where X and Y represent sets of unique k-mers identified as present in two different read sets. A

k-mer is defined as present if it’s frequency in a sample is ≥ 5. To get the genome-wide average

Jaccard dissimilarity ( ¯JD), we took the average of all the pairwise Jaccard dissimilarities.

Jaccard dissimilarity is likely the most commonly used k-mer-based diversity measure (Ondov

et al., 2016). However, whether a k-mer reaches the frequency threshold needed to be identified as

preset in a sample depends on the sequencing depth for the sample (VanWallendael and Alvarez,

2022). Thus, we also calculated Bray-Curtis dissimilarity (BD) between each pair of individuals

in a species’ dataset as:

BD(X, Y ) = 1 − 2

∑

i min(m∗
k
∑
i m∗

k

i (X), m∗

i (Y ))

i (X) + m∗

i (Y )

(2.4)

where m∗

i (X) gives the normalized frequency of k-mer i in genome X. The normalized frequencies

are calculated by taking each frequency mi(X) and dividing it by the sum of the raw frequencies

41

as in Dubinkina et al. (2016):

m∗

i (X) =

mi(X)∑

i mi(X)

(2.5)

This step accounts for variation in coverage between samples on k-mer frequency. To get the

genome-wide average Bray-Curtis dissimilarity ( ¯BD), we again took the average of all the pairwise

Bray-Curtis dissimilarities. Note that both Jaccard and Bray-Curtis dissimilarity are scaled in their

denominators by either the total number of unique k-mers or total number of k-mers respectively,

analogous to how nucleotide diversity is scaled by the number of sites included in the calculation.

Population size proxies

Following similar methods to Corbett-Detig et al. (2015) and Buffalo (2021), we defined cur-

rent census population size (N ) as the product of species range size (R) in square kilometers and

population density (D) in individuals per square kilometer:

N = RD

(2.6)

Estimation of both R and D are handled separately below. Importantly, these methods have

the same drawback as described in Corbett-Detig et al. (2015) and Buffalo (2021): contemporary

estimates of R and D do not necessarily reflect the historical values of R and D. However, since

nearly all the species in this study lack long-term historical data on their population size, it is not

currently possible to estimate long-term historical N without making strong assumptions.

Range size estimation from GBIF occurrence data

We first estimated range size based on Global Biodiversity Information Facility (GBIF) oc-

currence data from the rgbif package (Chamberlain and Boettiger, 2017). For each species, we

identified its GBIF taxon key(s). If the species is domesticated, we used the taxon key(s) for a wild

relative with an overlapping range when possible. We then downloaded all records associated with

each taxon key that had an occurrence status of “PRESENT”, had coordinates that mapped to land,

had any basis of record other than “FOSSIL SPECIMEN”, and recorded anywhere in a year ≥ 1943

42

and ≤ 2023. In addition, the records could not have any GBIF issue codes, except for the issue

codes listed in Appendix C. Similar to previous studies (Corbett-Detig et al., 2015; Buffalo, 2021),

we estimated range size for domesticated species using GBIF occurrences from closely-related wild

relatives because it is difficult to distinguish the native and introduced ranges of globally cultivated

crop species with only occurrence data. Note, however, that we also used an additional method

for estimating range size that is not burdened by this same assumption (see Range size estimation

from WCVP distribution maps). The relatives used for each domesticated species is detailed in

Table S1.

We followed methods of Buffalo (2021) to estimate range size from each species’ set of GBIF

occurrence data using the package alphahull (Pateiro-Lopez and Rodriguez-Casal, 2022). We

started with splitting the occurrence data by continent, in order to avoid estimating ranges that

overlapped with oceans. We also only kept occurrences with unique latitude-longitude values to

reduce the computational burden of alphahull’s algorithms. We then added a small amount of

random jitter (normally distributed with µ = 0 and σ = 1 × 10−3) to the latitude-longitude coordi-

nates of each unique occurrence to avoid errors in the triangulation algorithm of alphahull, which

can break when there are lots of colinear points. Finally, we filtered out any continents which had

fewer than 20 unique occurrences of a species. The only exceptions to this rule were Solanum

stenotomum, Dioscorea alata, and Rhododendron griersonianum, for which we only required 8, 6,

and 3 occurrences respectively due to the rarity of these species and thus a paucity of occurrence

data. We then used alphahull to compute the alpha shape of each continent subset, which can be

thought of as the smallest possible convex shape that encloses a set of points in a plane. We defined

the alpha parameter for the alphahull package to be 200. We then used the R packages sf (Pebesma,

2018) and rworldmap (South, 2011) to measure the sizes of the alpha shapes in square kilometers

after projecting them onto the Earth’s surface. Finally, we took the estimated range polygons and

filtered out ones that resided on continents in the introduced range of the species, as defined by the

World Checklist of Vascular Plants (WCVP) (Govaerts et al., 2021). The sum of the areas of the

remaining polygons was our estimate of range size.

43

Range size estimation from WCVP distribution maps

We also estimated range size from expert-drawn species distribution maps instead of species

occurrence data. We used the rWCVP package (Brown et al., 2023) to download distribution maps

from WCVP (Govaerts et al., 2021). We then estimated range size for each species as either (1) the

sum of the areas of all map elements labeled as “native” or “extinct” for that species or (2) the sum of

the areas of all map elements labeled as “native”, “invaded”, or “extinct” for that species. Regions

with an occurrence label of “dubious” were excluded from downstream analyses. In contrast to

GBIF-derived ranges, we used distribution maps for domesticated species in this estimate of range

size because the maps discriminate between the native and introduced ranges of species.

Population density estimation from plant height

Similarly to previous studies, we use plant height as a proxy for plant population density

(Corbett-Detig et al., 2015). While it would be ideal to use actual population densities in our anal-

yses, we could not find published estimates of population densities for many of the species in our

dataset and all previous studies investigating Lewontin’s paradox rely on population size proxies

(Leffler et al., 2012; Corbett-Detig et al., 2015; Filatov, 2019; Buffalo, 2021). We elaborate further

on the limitations of using proxies in the Discussion, but at the time of writing this manuscript using

proxies is the only way to achieve a sufficient sample size for investigating Lewontin’s paradox.

We decided to use plant height rather than plant mass (Deng et al., 2012) as our measure of body

size because plant height measurements are available for many more species in our dataset and also

to make our results more comparable to previous studies that also use plant height (Corbett-Detig

et al., 2015). According to theory outlined in Deng et al. (2012), where D is population density, M

is plant mass, and h is plant height, D ∝ M −3/4 and M ∝ h8/3. Combining these two relationships

gives D ∝ (h8/3)−3/4 which simplifies to D ∝ h−2. Adding this density-height relation to equation

2.6 gives our main proxy for population size:

N ∝ R
h2

44

(2.7)

In our subsequent analyses, we refer to Equation 2.7 as the range size-squared height ratio and

we convert R to square meters and h to meters to make the ratio unitless. As Equation 2.7 suggests,

we do not expect the range size-squared height ratio to exactly equal the true population size or be

interpretable as a number of individuals. Rather, it is a quantity we expect to scale with population

size. To calculate the range size-squared height ratio for each species, we downloaded plant height

data from the EOL, which mainly comprised records summarized from the TRY database. If no

height measurements were available for a species in the EOL, then we used estimates we found in

published scientific literature. The only exceptions to this were Vanilla planifolia and Rhododen-

dron griersonianum, where our height estimates came from the Kew Botanical Gardens’ and the

American Rhododendron Society’s websites, respectively. The sources used for each height value

are cited in Table S1.

Labeling species with genome size, mating system, ploidy, cultivation status, and life cycle
habit

Table S1 contains citations for all studies that were used to label each species in our study with a

genome size, mating system, ploidy level, cultivation status, and life-cycle habit. For determining

genome size, we used estimates from flow cytometry and k-mer-spectra analyses whenever possible

instead of using assembly size, since most assemblies do not contain the entire genome of the

sequenced species. Most of our genome size estimates were 1C values acquired from publications

cited in the Plant DNA C-values Database (Pellicer and Leitch, 2020). Any estimates in terms of

picograms (pg) of DNA were converted to base pairs using the following conversion factor: DNA

in Mb = DNA in pg ×0.978 × 109 (Doležel et al., 2003). If genome sizes in terms of pg were not

available for a species, then we used the size of the species’ genome assembly as the genome size.

We next labeled each species with a mating system (selfing, outcrossing, mixed, or clonal),

cultivation status (wild or cultivated), and life cycle habit (annual, biennial, perennial, or mixed)

because previous studies showed these factors to be important determinants of diversity in plants

(Chen et al., 2017). For classifying species into different mating systems, we used methods similar

to a previous study (Opedal et al., 2023) and generally considered species with outcrossing rate <

10 % as “selfing”, species with outcrossing rate between 10 - 90 % as “mixed”, and species with

45

outcrossing rate > 90 % as “outcrossing” when estimates of outcrossing rates were available. In the

absence of outcrossing rate data, we also labeled species described as generally self-incompatible

as “outcrossing” and we labeled species described as selfing as “selfing”. The only exception to this

was Oryza brachyantha for which we could not find mating system descriptions in peer-reviewed

literature. Thus, we assumed that this species was most likely outcrossing because most of the

other wild Oryza species in the dataset were classified as outcrossing. Because of the low number

of mixed (14) and clonal (2) species in our dataset, we collapsed the selfing, mixed, and clonal

species into a single “not outcrossing” category for later downstream analysis. Similarly, for life

cycle habit, our dataset contained only 1 biennial species and 2 species that had a mixture of annual,

biennial, and perennial forms. We combined these species with the perennial category to create a

single “not annual” category. For cultivation status, we looked up each species in the EOL and

classified species that had documented human uses (such as for food, fiber, fodder) or had some

countries known to cultivate the species as “cultivated”. All other species that did not meet these

criteria were classified as “wild”. The only exception to this was Lactuca sativa, which did not

have any human uses listed in EOL at the time of writing this paper; however, it is commonly

known as lettuce so we classified it as “cultivated”. Finally, for ploidy levels, when more than one

cytotype was described as present within a species we labeled the species with it’s most common

naturally-occurring cytotype. Citations to relevant literature used for each classification decision

can be found in Table S1.

Statistical analysis

The ultimate goal of our statistical analyses was to estimate the effect of our population size

proxies on measures of diversity, comparing the effects of using k-mer-based or nucleotide diver-

sity. To do this, we used the caper R package (Orme et al., 2018) to implement a statistical approach

similar to Whitney et al. (2010). We performed partial phylogenetic least squares regressions con-

trolling for evolutionary history (using a phylogeny obtained from timetree.org, Kumar et al. (2017,

2022)), mating system (outcrossing vs not outcrossing), cultivation status (wild vs cultivated), and

life cycle habit (annual vs not annual). We did not include sequencing coverage as a covariate at

46

this step because we would later exclude species with low coverage data from our analyses (see

Results) and the relationship between coverage and diversity saturates at higher levels of coverage

(Figures S3A, S3D, and S3G). We also did not include number of individuals as a covariate because

this did not correlate with any of our diversity measures (Figures S3C, S3F, and S3I). Similar to

Whitney et al. (2010), we also scaled the dependent variables to be unitless with a mean of zero and

unit variance across species (using the scale() function in R) before performing regression to make

slopes more comparable across models and account for the inherent differences in unit between

nucleotide and k-mer diversity metrics. This approach can be summarized as follows:

scale(diversity) = β0 + β1×log10(population size proxy) + β2×mating system + β3×cultivation

status + β4×life cycle habit + ϵ

where population size proxy refers to either Equation 2.7 or it’s components (range size and plant

height), diversity was estimated using either SNPs (log10(¯π)) or k-mers ( ¯JD or ¯BD), and the scale()

function performs a z-transformation to make diversity unitless with mean of zero and unit variance.

We also constructed a separate set of models where we included genome size as a covariate:

scale(diversity) = β0 + β1×log10(population size proxy) + β2×mating system + β3×cultivation

status + β4×life cycle habit + β5×log10(genome size) + ϵ

We controlled for genome size in a separate set of models because we had conflicting expecta-

tions on whether genome size would be a confounder or a mediator of the population size-diversity

relationship. In other words, the effect of population size on diversity could act through genome

size, since small populations may not experience strong enough selection to purge deleterious inser-

tions (Lynch and Conery, 2003). Including genome size as a covariate in this case would artificially

diminish the estimated effect of population size on diversity. Alternatively, genome size could fun-

damentally alter the mode of adaptation in plant species (Mei et al., 2018), making genome size a

confounder of the population size-diversity relationship.

After constructing our models, we visualized the relationship between population size and di-

versity or genome size and diversity with partial regression plots, following methods from Riddell

47

(1977) and Blomberg et al. (2012). Beginning with our initial phylogenetic least squares model:

y = Xβ + ϵ

(2.8)

where y is a vector of diversity values, X is the design matrix, β is a vector of regression coef-

ficients, and ϵ is a vector of residuals distributed normally about 0 with phylogenetic variance-

covariance matrix Ω. We first performed Cholesky decomposition on Ω to get matrix C such that:

Ω = CC T

(2.9)

We then took the inverse matrix C −1 and left-multiplied both sides of our regression equations

to get:

Which we will rewrite as:

C −1y = C −1Xβ + C −1ϵ

y∗ = X ∗β + ϵ∗

where y∗ = C −1y, X ∗ = C −1X, and ϵ∗ = C −1ϵ. In vector form, this equation is now:

y∗ = β0x∗

0 + β1x∗

1 + β2x∗

2 + ... + βn−1x∗

n−1 + ϵ∗

(2.10)

(2.11)

(2.12)

where β0x∗
0 is our intercept (Note that x0 was initially a column of 1’s before being transformed
by C −1). After fitting this model to our data with the standard lm() function in R, we collected

all terms besides the primary variable of interest, x∗

k (which would be a population size proxy or

genome size in our case), and subtracted them from both sides of the equation to get:

48

y∗ −

∑

i̸=k

βix∗

i = βkx∗

k + ϵ∗

(2.13)

We then plotted the values of x∗

k against y∗ −

∑

i̸=k βix∗

i , interpreting the slope (βk) as the effect

of the primary variable on the response, scaled for phylogenetic relationships and adjusted for the

effects of confounding factors.

Results

Low diversity species explained by low mean coverage

There were 112 species in our initial dataset, each with estimates of population size proxies,

nucleotide diversity, and k-mer diversity (Figure 2.1). Out of these 112 species, 102 were diploids,

9 were tetraploids, and one was hexaploid, with haploid genome sizes ranging from 105 Mb to 5.06

Gb (Table S1). These species were further broken down into 57 annual species vs 55 not annual

species (which were predominately perennial), 31 wild vs 81 cultivated species, and 55 outcrossing

vs 57 not outcrossing species (which were predominantly selfing). Species classified as annual

also tended to not be classified as outcrossing (χ2 = 18.9, p = 1.4 × 10−5, Figure C1C). However,

cultivation status was independent of both life cycle habit (χ2 = 4.07 × 10−31, p = 1, Figure C1A)

and mating system (χ2 = 0.53, p = 0.47, Figure C1B). The number of individuals sampled in each

species varied from 3 to 1200 and the average depth of sequencing per individual varied from 0.028x

to 79.7x (Figure C2). Variation in the depth of sequencing between individuals, quantified as the

coefficient of variation in base pairs sequenced, varied about 50-fold from 0.030 to 1.6 (Figure

C2). There were no missing values for any of the variables investigated in this study, but there

were three species with zero variant sites called (Capsicum annuum, Heliosperma pusillum, and

Papaver somniferum) because they had very low coverage sequencing datasets (average coverages

per individual of 0.035x, 0.169x, 0.043x, respectively, Table S1). We omitted these species from

all downstream analyses.

Before testing our central hypothesis, we investigated whether technical sequencing variables

could explain any of the diversity values observed in our dataset. Mean coverage correlated with

both nucleotide diversity (ρ = 0.33, p = 0.00033, Figure C3A) and k-mer diversity (Jaccard: ρ=

49

Figure 2.1: Our study includes 112 plant species across a wide range of population sizes and
diversity levels. Species labeled in purple were considered outliers and omitted from downstream
analyses (see Figure 2.2), but species labeled in green were retained. The phylogenetic tree is scaled
in millions of years. The WCVP full ratio is a unitless population size proxy equal to the ratio of
range area, estimated using WCVP range maps, to squared plant height and is log-transformed
(base 10, see Equation 2.7). Nucleotide diversity is genome-wide average diversity at four-fold
degenerate sites, log-transformed (base 10, see Equation 2.2). Capsicum annuum, Heliosperma
pusillum, and Papaver somniferum had nucleotide diversity values of zero and so have bars at the
plotting limit (log(0) = −∞). Bray-Curtis dissimilarity is average pairwise Bray-Curtis dissimilar-
ity across all pairs of individuals in a species’ sample (see Equation 2.4).

50

-0.53, p = 2.6×10−9, Figure C3D; Bray-Curtis: ρ = -0.34, p = 0.00021, Figure C3G). Coefficient of

variation in bp sequenced correlated strongly with k-mer diversity (Jaccard: ρ = 0.36, p = 0.00013,

Figure C3E; Bray-Curtis: ρ = 0.42, p = 4.7 × 10−6, Figure C3H) but not nucleotide diversity (ρ =

-0.088, p = 0.36, Figure C3B). The number of individuals sequenced did not correlate with either

nucleotide diversity or k-mer diversity (Figure C3C, C3F, C3I).

Given that mean coverage strongly correlated with both k-mer dissimilarity and nucleotide di-

versity, we decided to investigate this relationship further. We would expect that low coverage

sequencing would produce artificially low nucleotide diversity and artificially high k-mer diver-

sity. This is because our SNP-calling pipeline is generally tuned for high-coverage datasets (see

SNP calling pipelne), so low coverage datasets produce few confident SNP calls. Conversely, low

coverage sequencing only samples a small proportion of the total k-mer space in a set of reads.

Thus, two independent low coverage samples are unlikely to share many k-mers in common and

will have inflated k-mer diversity. Overall, we observed that species with very low mean coverage

had both low nucleotide and high k-mer diversity (Figure 2.2A, Figure C4A). In contrast, there

was no clear mapping of these abnormal diversity values with coefficient of variation in base pairs

sequenced (Figure C5) or the number of individuals sequenced (Figure C6). Based on these results,

we removed 10 species from our dataset with mean coverage per individual ≤ 0.5x. This included

three species (Capsicum annuum, Heliosperma pusillum, and Papaver somniferum) with zero vari-

ant sites called because they had very low coverage (0.035x, 0.169x, 0.043x, respectively, Table

S1). We also dropped 4 additional species with fewer than 1000 variant sites called: Camelina

sativa , Sesamum indicum, Setaria italica, Actinidia chinensis, the first of which were just slightly

above our coverage cutoff (0.63x, 0.92x, 0.63x, respecitively, Table S1). These filtering steps re-

moved many of the species with abnormally low diversity and high dissimilarity values excluding

(Figure 2.2B). In total, we kept data for 98 species for downstream hypothesis testing.

Range size-squared height ratio varies over more orders of magnitude than nucleotide
diversity

We next investigated whether Lewontin’s paradox applied to our dataset by comparing diversity

estimates against population size proxies. For each species, we estimated range size using either

51

Figure 2.2: Low coverage and low numbers of variant calls explains species with abnormally
low diversity. (A) shows the relationship between k-mer diversity and nucleotide diversity without
omitting species with ≤ 0.5x coverage or ≤ 1000 SNP calls. (B) shows the same relationship,
except species with ≤ 0.5x coverage or ≤ 1000 SNP calls are omitted. Each data point is a species.
All species’ points are colored by the log (base 10) of average genome-wide coverage per individual
for that species. Purple lines are loess smoothing lines with 95% confidence intervals shaded in
gray. Values across the top of each plot are Spearman correlation coefficients (ρ) and p-values that
test whether each correlation coefficient differs from zero.

52

GBIF occurrence data or WCVP range maps. Estimates from these two methods were significantly

correlated no matter whether invaded ranges (as defined in the WCVP range maps) were included

(ρ = 0.31, p = 0.00096, Figure C7A) or excluded (ρ = 0.48, p = 7.3×10−8, Figure C7B). The

omission of invaded ranges lowered the range size of several plant species based on WCVP range

maps (Figure C7C) but had less effect on ranges estimated from GBIF occurrence data (Figure

C7D).

We then calculated the ratio of range size to squared plant height (Equation 2.7) using height

values from the EOL. We used this ratio as our primary population size proxy in downstream anal-

yses. After excluding species with < 0.5x coverage and < 1000 variant sites called (Figure 2.2),

nucleotide diversity varied over about 4 orders of magnitude for the species in our dataset (from

0.00021 to 0.117, Table S2), while the ratio of range size to squared plant height based on WCVP

and GBIF range estimation methods (including both native and invaded ranges) varied over 10

(from 8.9 × 108 to 1.7 × 1018) and 13 (from 8.6 × 105 to 1.5 × 1018) orders of magnitude, re-

spectively (Table S2). Mean pairwise Bray-Curtis dissimilarity values varied about 4.9-fold across

species, from 0.13 to 0.64, while mean pairwise Jaccard dissimilarity varied about 22-fold, from

0.040 to 0.87 (Table S2). Bray-Curtis dissimilarity values correlated with Jaccard dissimilarity

values across species (ρ = 0.76, p < 2.2 × 10−16, Figure C8).

k-mer diversity scales with population size proxies more than nucleotide diversity

The core of Lewontin’s paradox is that a population’s diversity does not scale much with popu-

lation size. If k-mers capture a wider range of genetic variation compared to SNPs, population size

will scale more with k-mer diversity than nucleotide diversity. If we did not control for shared evo-

lutionary history or any confounding variables (mating system, life cycle habit, cultivation status,

or genome size), then none of our diversity measures significantly correlated with the range size-

squared height ratio (Figure C9). After controlling for confounding variables, nucleotide diversity

marginally scaled with the range size-squared height ratio (β = 0.14, SE = 0.056, p = 0.017, Fig-

ure C10A). However, the relationship between k-mer diversity and the range size-squared height

ratio was highly significant, with generally a greater slope (Jaccard: β = 0.64, SE = 0.096 p =

53

2.2 × 10−9, Figure C10B; Bray-Curtis dissimilarity: β =0.79, SE = 0.11, p = 7.3 × 10−11, Figure

C10C). We observed the same qualitative trend when we included both native and invaded ranges

in the range size-squared height ratio (Figure C10D-F), or used the GBIF-based range estimates

instead of WCVP-based estimates (Figure C11). Interestingly, we often observed Bray-Curtis dis-

similarity having a larger slope with the range size-squared height ratio compared to Jaccard dis-

similarity (β = 0.64 vs 0.79 Figure C10B-C), but models where Bray-Curtis dissimilarity was the

response variable generally had lower adjusted R2 (e.g. R2 = 0.51 vs 0.41, models 2 and 3 in Table

S4).

We also analyzed range size and plant height separately as population size proxies (Figure C12-

C14). Overall, WCVP-estimated range size significantly affected nucleotide diversity (β = 0.29,

SE = 0.072, p = 0.00011, Figure C12A) and k-mer diversity (Jaccard: β = 0.92, SE = 0.13, p

= 9.9×10−11, Figure C12B; Bray-Curtis: β = 1.2, SE = 0.13,p = 3.2×10−14, Figure C12C), and

this trend held when we estimated range size from GBIF occurrences (Figure C13A-C) or included

invaded range area (Figure C12D-F and Figure C13D-F). On the other hand, nucleotide diversity

did not scale with plant height (β = 0.13, SE = 0.19, p = 0.5, Figure C14A), but k-mer diversity

marginally scaled downward with plant height (Jaccard: β = −0.78, SE =0.38, p = 0.046, Figure

14B; Bray-Curtis: β = −0.77, SE = 0.44, p = 0.088, Figure C14C).

Finally, we repeated our partial phylogenetic regressions controlling for genome size as an

additional covariate. In this case, nucleotide diversity did not scale with the range size-squared

height ratio (β = 0.035, SE = 0.063, p = 0.58, Figure 2.3A), but k-mer diversity did (Jaccard:

β = 0.54, SE = 0.093, p = 8.8×10−8, Figure C15; Bray-Curtis: β = 0.7, SE = 0.098, p = 2.2×10−10,

Figure 2.3B). Again, we got qualitatively similar results when we excluded invaded ranges in our

range size estimates (Figure C16), used GBIF occurrences to estimate range size-squared height

ratio (Figure C17) or used WCVP range size as the population size proxy (Figure C18). However,

GBIF range size by itself did not scale with Jaccard dissimilarity (Figure C19B, C19E). Increased

plant height associated with decreased k-mer diversity, but had no significant relationship with

nucleotide diversity (Figure C20).

54

Figure 2.3: k-mer diversity scales with population size proxies after controlling for genome
size, life cycle habit, mating system, and cultivation status. Purple lines are partial phylogenetic
regression lines between diversity levels and the population size proxy. The y-axis (diversity)
is scaled and adjusted according to Equation 2.13: scaling diversity levels to a standard normal
distribution, followed by correcting for phylogenetic relatedness and adjusting for the confounding
variables (genome size, life cycle habit, mating system, and cultivation status). The values at the
top of each plot give the slope of the partial regression ± one standard error and p-values testing
whether the slopes differ from zero. Dotted lines show the partial regression slope ± one standard
error. WCVP full ratio is a population size proxy estimated as the ratio of range size recorded in
WCVP range maps (including invaded ranges) to squared plant height.

55

k-mer diversity scales with genome size more than nucleotide diversity

We also investigated the relationship between diversity and genome size because we expected

genome size to potentially play a role in the mechanism underlying the greater scaling of k-mer

diversity with population size. Genome size is often a strong predictor of diversity (Lynch and

Conery, 2003). Among eukaryotes, variation in genome size is largely explained by variation in

transposable element abundance (Flavell et al., 1974; Kidwell, 2002; Lynch and Conery, 2003;

Muñoz-Diez et al., 2012; Tenaillon et al., 2011; Nystedt et al., 2013; Ibarra-Laclette et al., 2013),

which contribute substantially to the repetitive sequence content of genomes and increase the dif-

ficulty of aligning short reads to a reference genome (reviewed in Goerner-Potvin and Bourque

(2018)). Thus, our expectation was that k-mer-based diversity measures are more sensitive to

genome size variation compared to nucleotide diversity.

Increasing genome size was associated with decreasing k-mer diversity (Jaccard: β = −3.7,

SE = 0.42, p = 8.4×10−14, Figure C21; Bray-Curtis: β = −4.2, SE = 0.45, p = 4.5×10−15, Figure

2.4B) and nucleotide diversity (β −1.8, SE = 0.29, p = 1.4×10−8, Figure 2.4A), after controlling for

variation in the range size-squared height ratio, mating system, life cycle habit, cultivation status,

and evolutionary history. We got qualitatively similar results when the population size proxy we

corrected for excluded invaded ranges (Figure C22), or if our population size proxy was based on

GBIF occurrences (Figure C23), or we used range size or plant height individually to control for

population size variation (Figure C24-C26). Across all of these analyses, the partial regression

relationship between genome size and diversity was always significantly negative.

Discussion

Our goal was to investigate whether genomic approaches that can capture more genetic vari-

ation than reference-based methods can improve the scaling between population size proxies and

diversity. This was motivated by literature suggesting suggesting SNPs called against a single ref-

erence provide an incomplete picture of genome-wide polymorphism (Schmidt et al., 2021; Van-

Wallendael and Alvarez, 2022; Jaegle et al., 2023; Sopniewski and Catullo, 2024). In total, we

processed >205 terabases of publicly available sequencing data from the SRA over approximately

56

Figure 2.4: k-mer diversity is more sensitive to genome size variation than nucleotide diversity.
Purple lines are partial phylogenetic regression lines between diversity levels and genome size (see
Equation 2.13) after scaling diversity levels to a standard normal distribution (mean = 0, variance
= 1), followed by scaling diversity levels and population sizes according to their phylogenetic
relatedness, and finally adjusting for the confounding effects of mating system, cultivation status,
life cycle habit, and population size. Here we used the ratio of range size to squared plant height,
where range size was estimated from ranges in WCVP range maps (including invaded ranges). The
values at the top of each plot give the slope of the partial regression ± one standard error and p-
values testing whether the slopes differ from zero. Dotted lines show the partial regression slope
± one standard error.

57

12 months of wall time, split between a maximum of 512 cores and 50 TB of disk space. After

careful accounting for potential technical and phylogenetic confounding, the standardized slope

between k-mer-based diversity and the range size-squared height ratio was up to 20 times larger

than the same standardized slope for nucleotide diversity (β = 0.035 vs 0.7, Figure 2.3). We

observed similar results across the two different measures of range size (Figure C17) and k-mer

diversity (Figure C15). We also observed that k-mer-based diversity is more sensitive to variation

in genome size compared to nucleotide diversity (Figure 2.4). Overall, these results are consistent

with the possibility that diversity missed by reference-based analyses or changes in the diversity

calculation itself can partly explain the weak scaling between population size and diversity. How-

ever, a larger pangenome scale analysis will be required to fully understand the genetic variants

underlying this result.

Extending our findings to Lewontin’s paradox directly is complicated by a few factors. First, it

would be ideal to compare our diversity estimates to neutral expectations of diversity vs population

size. For SNPs, the expected diversity under neutrality would be 4Neµ or a similar formula. For

k-mers the exact value of µ is not clear, as k-mers also reflect non-SNP variation. Several k-mer-

based measures of diversity are nearly perfectly correlated with π in neutral models where only

SNPs are considered (Roberts et al., 2024), but the inclusion of non-SNP variation would probably

change this relationship. Identifying the concrete variants underlying k-mers would help us better

understand µ for k-mer, but this is currently not possible for mutations other than SNPs and small

indels without pangenomic references (Uricaru et al., 2015; Gauthier et al., 2020). The eventual

release of pangenomic references across many species will provide better context for increased

scaling of k-mer diversity with population size. Similar to previous Lewontin’s paradox studies,

we also use both non-coding sequences (for k-mers) and 4-fold degenerate sites (for SNPs) as

standards for estimating neutral diversity (Leffler et al., 2012; Buffalo, 2021), but there could be

differences between coding and non-coding values of µ that contribute to results observed in our

work and previous work. More estimates of µ are needed for non-SNP mutations (Quiroz et al.,

2023), but we further caution that many estimates of µ will be limited in that they usually reflect

58

contemporary populations. Variation in µ over time, such as through TE bursts (Belyayev, 2014),

could play an important role in diversity vs population size scaling and warrant further research.

In addition to limitations estimating µ and π, estimates of Ne also tend to be limited in inves-

tigations of Lewontin’s paradox. Like previous analyses, our analysis assumes that contemporary

population size estimates are good proxies for historic population sizes (Corbett-Detig et al., 2015;

Buffalo and Coop, 2020). Common population size proxies such as range size and plant height

only reflect the current census population size of a species. However, it is the long-term harmonic

mean of the effective population size, not just the current size, that determines diversity levels

within a population (Wright, 1940). There is some literature correlating aspects of species history

with contemporary diversity levels (López￿Delgado and Meirmans, 2022) and incorporating these

metrics into Lewontin’s paradox is an exciting avenue for future research. As a first step in this

direction, we used plant range maps into native and invaded ranges to test the robustness of our

results to invasion-related range size changes (Brown et al., 2023). Overall, our observations were

remarkably similar no matter whether we included or excluded invaded ranges in our population

size proxies (Figure C17A-C vs Figure C17D-F). Part of this apparent robustness was due to the

insensitivity of our GBIF-based range size estimates to the inclusion of invaded ranges (Figure

C7D). However, our WCVP-based range size estimates were drastically altered by the inclusion

of invaded ranges (Figure C7C) and still yielded similar results (Figure 2.3, C15, C16). Although

we cannot rule out the possibility that older historical events have affected contemporary diversity

levels, our results appear to be robust to recent biological invasions.

As with all regression-based analyses, our results are also ultimately sensitive to error in the

measurement of both covariates (population size proxies, genome size, mating system, life cycle

habit, or cultivation status) and outcome variables (nucleotide or k-mer diversity). For example,

we did not filter out potentially contaminating sequences from our k-mer analysis, which could add

noise to our k-mer dissimilarity values. However, this could only explain the increased scaling be-

tween k-mer diversity and population size if plants with larger population size proxies had a higher

diversity of contaminants, which seems unlikely given that many of the sequencing datasets in our

59

study are from laboratory cultivated plants (see Table S1) and that we excluded low-frequency k-

mers from our analyses. Our study is also unique in the multiple steps we took to limit the influence

of systematic measurement errors on our coefficients, including reanalyzing all population-level

sequencing data with one pipeline to limit the impact of bioinformatic parameter choices on our

analysis (Mirchandani et al., 2024), filtering out low coverage datasets(Sandell et al., 2022), ac-

counting for missing data in calculations of nucleotide diversity (Schmidt et al., 2021; Korunes

and Samuk, 2021)and estimating range size with two different methods (WCVP range maps and

GBIF occurrence records, Figure C6). Although we could not control for some covariates (Willis,

1922; Romiguier et al., 2014; Guo et al., 2024) due to a dearth of data, our study is still the largest

reanalysis of population-level sequencing data in plants that we know of to date. The availability of

our workflow also makes it easy for our study to be extended as more population-level sequencing

data is released.

Interestingly, the estimated effect of our population size proxies on diversity was often slightly

larger for Bray-Curtis dissimilarity than Jaccard dissimilarity (for example, β = 0.7 vs 0.54 from

Figure 2.3B vs Figure C15, Table S4). In contrast, the range size-squared height ratio was of-

ten slightly more predictive of Jaccard dissimilarity than Bray-Curtis dissimilarity (Table S4). We

could not test whether these trends were statistically significant, but the benefits of different k-mer

metrics in predicting measures of population size warrant further study. Our expectation is that

k-mer diversity measures based on frequency, such as Bray-Curtis dissimilarity, better capture di-

versity compared to measures based on purely k-mer presence/absence, such as Jaccard dissimilar-

ity, because they explicitly measure copy number variation. However, accurately measuring k-mer

frequencies likely requires higher sequencing coverage than calling presence/absence, which could

explain why Bray-Curtis dissimilarity generally scaled more with population size but had a lower

R2 compared to Jaccard dissimilarity (Table S4). Future studies using higher coverage population

level sequencing data could help test this hypothesis.

k-mer frequencies are known to be highly informative of genomic structure, with one common

application of k-mers being the estimation of genome size (Vurture et al., 2017; Pflug et al., 2020).

60

Similar to previous studies, we observed that nucleotide diversity was negatively correlated with

genome size (Lynch and Conery, 2003; Chen et al., 2017), but we observed an even stronger neg-

ative correlation for k-mer diversity (β = −1.8, SE = 0.29 vs β = -3.7, SE = 0.42 in Figure 2.4).

k-mers also appeared to explain diversity patterns that scaled with population size beyond those

explained by genome size, while nucleotide diversity did not. After controlling for genome size,

the relationship between our population size proxies and nucleotide diversity was not significant

(Figure 2.3A, C17-C19 panels A and D), but the relationship between k-mer diversity and popula-

tion size proxies was often still highly significant (Figure 2.3B, C17-C19 panels B; C; E; F). The

only exception was that Jaccard dissimilarity did not significantly scale with GBIF-based estimates

of range size (Figure C19B, C19E). This additional scaling of k-mer diversity with population size

beyond just the effects of genome size and confounding variables suggests that k-mers capture

some element of the population size-diversity relationship that is absent from nucleotide diversity.

Our results do not negate the fact that other important factors also underlie Lewontin’s paradox,

such as past demographic fluctuations and linked selection. However, our results do suggest that

future studies of Lewontin’s paradox may want to consider diversity outside one reference genome.

The increasing availability of pangenomes across species (Göktay et al., 2021; Zhou et al., 2022;

Rice et al., 2023; Wang et al., 2023) offers many opportunities to revisit this classic population

genetics question. Ideal future studies would use pangenomic genotyping methods across a wide

range of species with a standardized pipeline, combined with multiple proxies of population size,

and understandings of each species’ demographic history. Altogether, these methodological devel-

opments will hopefully reveal a more wholistic picture of variation across the tree of life.

Data availability

Supplemental figures are in appendix C. Our entire analysis is packaged as a snakemake work-

flow stored here: https://github.com/milesroberts-123/tajimasDacrossSpecies. All supplementary

tables will be made available online at Evolution Letters. Table S1 contains the metadata for all

of the datasets used in this study, including sources for genome assemblies, genome annotations,

population-level sequencing datasets, and GBIF observations. Table S2 contains all of the covari-

61

ate and response variable values used for fitting our phylogenetic least squares models. Table S3

contains the estimated coefficients of all of our phylogenetic least squares models and their related

statistics, including p-values and standard errors. Table S4 contains the model-level statistics for

each phylogenetic least squares model, including R2 values and F-test results.

62

CHAPTER 3: k-MER-BASED APPROACHES TO BRIDGING PANGENOMICS AND
POPULATION GENETICS3

Abstract

Many commonly studied species now have more than one chromosome-scale genome assembly,

revealing a large amount of genetic diversity previously missed by approaches that map short reads

to a single reference. However, many species still lack multiple reference genomes and correctly

aligning references to build pangenomes can be challenging for many species, limiting our ability to

study this missing genomic variation in population genetics. Here, we argue that k-mers are a very

useful but underutilized tool for bridging the reference-focused paradigms of population genetics

with the reference-free paradigms of pangenomics. We review current literature on the uses of

k-mers for performing three core components of most population genetics analyses: identifying,

measuring, and explaining patterns of genetic variation. We also demonstrate how different k-

mer-based measures of genetic variation behave in population genetic simulations according to the

choice of k, depth of sequencing coverage, and degree of data compression. Overall, we find that

k-mer-based measures of genetic diversity scale consistently with pairwise nucleotide diversity (π)

up to values of about π = 0.025 (R2 = 0.97) for neutrally evolving populations. For populations

with even more variation, using shorter k-mers will maintain the scalability up to at least π = 0.1.

Furthermore, in our simulated populations, k-mer dissimilarity values can be reliably approximated

from counting bloom filters, highlighting a potential avenue to decreasing the memory burden of k-

mer based genomic dissimilarity analyses. For future studies, there is a great opportunity to further

develop methods to identifying selected loci using k-mers.

Introduction

Two decades ago, assembling one reference genome for one eukaryotic species was an inter-

national, herculean effort (Venter et al., 2001). Now, individual laboratories can readily assemble

and align multiple reference-quality genomes from the same species into pangenomes (Golicz et al.,

2020). This shift toward pangenomes as the basis for genetic studies is already transforming our

3This chaper is published at the following DOI under a Creative Commons CC BY License: https://doi.org/10.
1093/molbev/msaf047. This license permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited. Permission to reuse this article is not required.

63

understanding of genetic variation in populations. Analysis of pangenomes has uncovered vast

quantities of genetic variation previously missed by the ubiquitous practice of aligning short reads

to a single reference genome (Wong et al., 2020; Sirén et al., 2021; Ebler et al., 2022; Zhou et al.,

2022; Rice et al., 2023; Liao et al., 2023), increased the power of trait mapping (Song et al., 2020;

Chin et al., 2023), and resolved complex structural variations (Hickey et al., 2020; Song et al.,

2024). Pangenomes have even revised our understanding of previously cataloged variation, be-

cause single nucleotide polymorphisms (SNPs) once identified by mapping reads against a single

reference can sometimes be resolved as alignment errors due to structural variation (Jaegle et al.,

2023). Altogether, pangenomes provide a more accurate representation of genetic variation, re-

ducing the commonly observed phenomenon where one’s choice of reference genome shapes the

ultimate conclusions of a study (i.e. reference bias, Gage et al. (2019); Günther and Nettelblad

(2019); Chen et al. (2021); Ebler et al. (2022)). This better ability to capture and explain pat-

terns of genetic variation, combined with recent developments in pangenome assembly algorithms

(Garrison et al., 2024; Hickey et al., 2020) and an explosion in pangenome sequencing for many

non-model organisms (Lei et al., 2021), means that pangenomes will likely be the standard for

population genetics analysis in the near future.

However, pangenomes can be difficult to assemble and tune in some contexts (Hahn, 2018;

Song et al., 2024). First, pangenomes by definition require more sequencing data to assemble

compared to a single reference genome, which can make building pangenomes expensive for large

genome species or study systems with fewer resources. Furthermore, whole genome alignment

(WGA) is key for pangenome assembly but also difficult to tune, affecting pangenome analysis.

Modern WGA algorithms are impressive and are being used to produce pangenomes for a wide

range of species (Garrison et al., 2024); however, current WGA algorithms are mainly developed

and tuned to align human and model species genomes, and so do not always generalize well to

genomes that are large, highly repetitive, highly diverse, polyploid, or containing high levels of

structural variation (reviewed in Song et al. (2024)). For example, corn (Zea mays) has a famously

repetitive and structurally variable genome and new WGA approaches needed to be developed just

64

to properly align corn genomes (Song et al., 2022). There are also still many open questions on

how to best represent complex, nested variations in such alignments and tune alignment parameters

(Song et al., 2024). A researcher’s exact choice of alignment parameters and software can dras-

tically affect the shape of a pangenome graph (Rice et al., 2023) and downstream genotype calls

(O’Rawe et al., 2013; Li, 2014; Bush et al., 2020; Betschart et al., 2022; Sopniewski and Catullo,

2024). Given the challenges of computing and tuning alignments, approaches that skip alignment

altogether could valuably complement or help guide pangenome assembly.

In this review we argue that the k-mers deserve more attention from population geneticists be-

cause they can complement the study of pangenomes. A k-mer is a sub-sequence of length k within

a larger sequence. For example, the sequence “ATGCA”, contains the unique 2-mers AT, TG, GC,

and CA. The main benefit of k-mers is that they can be analyzed without alignment. Instead, one

simply counts all of the unique k-mers present in a sample of reads (where k-mers that are reverse

complements of each other are typically considered identical and are counted together Kokot et al.

(2017)), then uses the resulting count matrix for downstream analysis (see Figure 3.2). Thus, k-

mers derived from regions far diverged, absent, or otherwise unalignable in a given reference will

not be automatically excluded from an analysis, allowing one to get a picture of pangenomic vari-

ation. k-mers have a long history of use in metagenomics (McClelland, 1985; Rosen et al., 2008;

Dubinkina et al., 2016), phylogenetics (Kolekar et al., 2012; Haubold, 2014; Zielezinski et al., 2017,

2019; Bussi et al., 2021; Beichman et al., 2023; Jenike et al., 2024), computer science (Shannon,

1948), and quantitative genetics (Voichek and Weigel, 2020; Kim et al., 2020; Gupta, 2021; Onetto

et al., 2022; Lemane et al., 2022). However, while applications of k-mers receive much attention

in other disciplines, population genetic investigations using k-mers remain limited.

Our goal is to review k-mer-based approaches for identifying, measuring, and explaining pat-

terns of genetic variation in populations. At the same time, we investigate the behavior of k-mer-

based measures of variation to the choice of k, the depth of sequencing coverage, and the degree of

data compression. We finally highlight some avenues to explore in k-mer-based (i.e. reference-free

or alignment-free) population genetics. Overall, we advocate that k-mer-based approaches can be

65

valuable complements to common reference-based population genetics methods.

Box 1: What is the “best” value for k?

A common first question in k-mer based analyses is: What value(s) of k should be analyzed?

The “best” k for an analysis is ultimately determined by a trade-off between the length of k and

sequencing error: longer k-mers are more likely to represent unique genomic sequences, but are

also more likely to contain a sequencing error (Rahman et al., 2018). In practice, many studies use a

k of around 20-40 bp (Ponsero et al., 2023) because k-mers in this range can be reliably sequenced

with short read data and often align uniquely to their source genome (Wu et al., 1991; Becher et al.,

2022). For example, k = 32 captures 85.7 % of unique sequences in the human genome (Shajii

et al., 2016), while k = 21 distinguishes many eukaryote, bacteria, and archea species (Bussi et al.,

2021). However, there are many past studies that propose criteria for choosing specific values of

k.

Choosing multiple values of k

One “brute force” approach to test the sensitivity of results to k is to simply repeat an analysis

multiple times for different values of k. This approach is especially common among genome as-

sembly algorithms (Chikhi and Medvedev, 2014; Durai and Schulz, 2016) but could be applied to

almost any analysis in theory. The main drawback, however, is the high computational burden of

performing the same analysis multiple times. It is also difficult to know without more information

whether analyses performed for certain values of k produce more accurate results than analyses

for other values of k, motivating the need for k selection criteria that can be either minimized or

maximized.

Choosing k based on the number of unique non-erroneous k-mers

Higher values of k generally allow greater detection of unique sequences, but increase the prob-

ability of observing k-mers containing at least one sequencing error (Chikhi and Medvedev, 2014;

Rahman et al., 2018). Each sequencing error can result in up to k erroneous k-mers, making it

important to prevent errors from dominating one’s analysis. Thus, choosing a k that maximizes the

number of unique non-erroneous k-mers in a dataset is generally considered optimal for tasks like

66

genome assembly (Chikhi and Medvedev, 2014). This approach generally involves measuring the

k-mer frequency spectrum and then fitting a model to the distribution to estimate which parts of the

spectrum come from erroneous k-mers (Chikhi and Medvedev, 2014). Usually, the low-frequency

end of the spectrum is dominated by erroneous k-mers because sequencing errors are unlikely to

generate the same erroneous k-mers many times and usually convert real k-mers into k-mers not

found in the source genome (Kelley et al., 2010). Although this criterion for choosing k could be

applied to population genetic datasets, it is not if clear the resulting optimal k would vary signifi-

cantly between genomes within the same species. Presumably, if genomes within the same species

have considerable variation in repetitive content (Haberer et al., 2020), size (Schmuths et al., 2004),

or ploidy (reviewed in Kolář et al. (2017)) then the optimal k could vary. Determining the value of

k that maximizes the number of unique non-erroneous k-mers across all individuals in a population

may be of interest for future population genetics studies.

Choosing k based on the probability of chance k-mer matches between samples

Another way to choose k is to think about the probability that a k-mer from one genome is also

found in a second genome by chance alone. For example, it would be unsurprising for almost any

pair of reasonably long, naturally-occurring DNA sequences to share the k-mer “AGC” because

this k-mer is very short and could occur many times in a random sequence. How large must k be

then before finding a k-mer in two different sequences is unlikely by chance alone? To discuss this

question, we next present equations similar to ones in Ondov et al. (2016), except we generalize to

account for variation in base composition between sequences being compared.

Let’s begin by imagining we have two genome sequences we wish to compare: X1 and X2.

First, to generate X1, we sample with replacement the letters A, T, G, and C a total of L times with

probabilities pA1, pT 1, pG1, and pC1, respectively where pA1 + pT 1 + pG1 + pC1 = 1. In other words,

we sample our DNA from a multinomial distribution where we assume that each base is sampled

independently of the preceding bases. This gives us one strand for X1 and we can then generate

the complementary strand by pairing A with T and G with C. We repeat this whole process once

more to construct X2 using probabilities pA2, pT 2, pG2, and pC2, which may or may not match the

67

probabilities for X1, except we only sample k bases. This gives us a k-mer, K, in the genome

sequence of X2. We now wish to find the probability that K occurs in X1, assuming that the

sampling of bases for K was independent of sampling bases for X1.

If we imagine constructing X1 and K at the same time, then the probability of drawing the same

letter for both sequences at a given position is:

ΣF = pA1pA2 + pT 1pT 2 + pG1pG2 + pC1pC2

(3.1)

However, we also want to consider sequences that are reverse complements as identical, so the

probability of drawing a pair of bases that match as reverse complements is:

ΣR = pA1pT 2 + pT 1pA2 + pG1pC2 + pC1pG2

(3.2)

The probability of drawing k pairs of bases in a row that are either identical or reverse comple-

ment matches is thus Σk

F and Σk

R, respectively, assuming that each base is sampled independently

of all others. The complementary probability of not getting k matches in a row on the forward

strand is 1 − Σk

F and, comparably, 1 − Σk

R for the reverse strand. Because X1 contains L bases,
there are a total of L − k + 1 k-mers in X1, so the probability of a K not matching at any k-mers
R)L−k+1. It should be noted that at this step

in X1 is approximately (1 − Σk

F )L−k+1 and (1 − Σk

we are assuming that the chance of K mismatching at a given position in X1 is independent of

the chance that K mismatches at other positions in X1, which is not strictly true. For example, if

K is the sequence “AAAAA” and we compare K to a subsequence in X1 that is “AACAA” we

would say that K and this subsequence do not match. However, with this information we would

also know with certainty that adjacent k-mers in X1 will also not match K because they will still

contain a “C”. However, we will continue with assuming that the mismatch between X1 and K is

independent at all positions because this makes our final equations more conservative (decreasing

the Σk

F term in Equation 3.5 will decrease the overall value of Equation 3.5).

We can now say that the probability of finding the k-mer K at least once by chance alone at

68

any of the L − k + 1 positions in the sequence X1 is:

P (K ∈ X1F ) = 1 −

P (K ∈ X1R) = 1 −

(

(

)(L−k+1)

1 − Σk
F

)(L−k+1)

1 − Σk
R

Assuming k << L, Equations 3.3 and 3.4 approximate to:

P (K ∈ X1F ) ≈ 1 −

P (K ∈ X1R) ≈ 1 −

(

(

)L

1 − Σk
F

)L

1 − Σk
R

(3.3)

(3.4)

(3.5)

(3.6)

We confirmed that equations 3.5 and 3.6 work as expected in simulations (Figure 3.1). Similar

to Fofanov et al. (2004), we can then solve these equations to give the minimum k-mer length

required to achieve a desired probability of chance k-mer matching of q = P (K ∈ X1F ) =

P (K ∈ X1R):

⌈

(

kF =

logΣF

1 − (1 − q)1/L

⌈

(

kR =

logΣR

1 − (1 − q)1/L

)⌉

)⌉

(3.7)

(3.8)

and now given a choice of q we are willing to tolerate, we can then use max(kF , kR) as a potential

choice of k.

Equations 3.7 and 3.8 demonstrates that k = 19 reduces the probability of two 3 Gb genomes

of random sequence sharing the same k-mer by chance to just 1 % (Ondov et al., 2016) (assuming

69

all bases are in equal proportion in both genomes, pA1 = pT 1 = pG1 = pC1 = pA2 = pT 2 = pG2 =

pC2 = 1/4) while k = 27 gives q = 1×10−6 for a 10 Gb genome where both X1 and X2 have a GC

conent of 42 % (pA1 = pT 1 = pA2 = pT 2 = 0.29 and pG2 = pC2 = pG2 = pC2 = 0.21). Slightly

longer k-mers are required when the proportion of each base is not 25 % in both genomes because

it is more likely then to have low-complexity strings of bases which make spurious matches more

likely. For example, the required k-mer length increases to 34 bp if the GC content of both genomes

is 21 % (pA1 = pT 1 = pA2 = pT 2 = 0.395 and pG1 = pC1 = pG2 = pC2 = 0.105), given a 10 Gb

genome and q = 1 × 10−6.

In summary, if k is high enough, two genomes in a population are highly unlikely to share

k-mers just through the accumulation of random sequences alone, suggesting that shared k-mers

usually reflect shared ancestry. Developing additional k-mer selection criteria that explicitly model

the influence of shared ancestry on the probability of k-mer sharing could improve approaches to

choosing k.

70

Figure 3.1: The relationship of k-mer length and base composition to chance matching. We
verified (A) Equation 3.5 and (B) Equation 3.6 by randomly generating 5.5 million pairs of k-mers
and genomes. Each simulation had 100,000 trials. For each trial we randomly generated a genome
of length 10,000 bases using a given base composition (pA, pT , pG, pC in barchart) and also gener-
ated a random k-mer (color gradient shows k) using the same base composition. We then checked
whether the random k-mer was in the random genome. In total, the 5 different base compositions
× 11 different k-mer lengths × 100,000 trials per simulation gives 5.5 million trials total. Grey
line is where the theoretical (i.e. Equations 3.5 and 3.6) and empirical probabilities (i.e. from sim-
ulations) of a chance match are equal. Dotted black line shows the resolution of the simulations;
because we did 100,000 trials per simulation we cannot empirically estimate probabilities smaller
than 1/100,000.

Choosing k based on the balance of shared vs differing k-mers

A final class of approaches for choosing k focuses on how if k is small, most k-mers are shared

between most samples, but as k increases more sample-specific k-mers occur until no shared k-

mers remain (Zhang et al., 2017a). Neither of these scenarios is usually desirable - the former

makes all samples appear to be identical, while the latter makes all samples appear to be completely

different. Thus, choosing k boils down to balancing the number of shared vs differing k-mers. Some

statistics for informing this choice include cumulative relative entropy (Sims et al., 2009), relative

sequence divergence (Sims et al., 2009), average number of common features (Zhang et al., 2017a),

Shannon’s diversity index (Zhang et al., 2017a), and χ2 tests (Bai et al., 2017). However, there are

two main drawbacks of these statistics. First, they were mainly developed for phylogenetic studies

focused on short amino acid k-mers, so their utility for population genetics (which would probably

focus on longer nucleotide k-mers) is untested. Furthermore, different values of k will appear

71

optimal for genomes of different sizes (Zhang et al., 2017a), making it hard to imagine that one

optimal k exists for a population exhibiting substantial genome size variation. Extending these or

similar approaches to optimize k for populations with genome size variation will be very useful for

future studies.

Box 2: What’s the expected value of k-mer diversity in a neutrally evolving population?

We argue that k-mers can be useful for initial assessments of genetic diversity, but two key

questions are likely of interest to population geneticists: can k-mer diversity be related to nucleotide

diversity (the common measure of genetic diversity) and what is the expected value of k-mer metrics

for populations evolving under a given model? Here, we derive a simple bound on the expected

number of k-mer differences between a pair of individuals in a neutrally evolving population. Our

result is similar to (Shi et al., 2024), except we generalize beyond haploid organisms.

We start by assuming that (1) k is sufficiently long to capture all of the unique sequences in a

genome, (2) each genome is sequenced at sufficient coverage to confidently identify all of the k-

mers present, (3) k-mers are sequenced without error, and (4) only SNPs contribute to differences

between genomes. Let i and j be individuals in a neutrally evolving population with ploidy level

x and let y and z be two haploid genomes in the pool of i and j that differ by origin. Next, let Ki

and Kj represent the set of k-mers identified in individual i and j’s genomes respectively and let

|Ki| denote the size of the set Ki. If all of the genomes within i and j are identical, then all k-mers

are shared between them. If we start by assuming i and j are haploid, this means a maximum of 2k

k-mers are not shared between i and j for every pairwise difference between i and j (Iqbal et al.,

2012; Younsi and MacLean, 2015). This relationship can be written as:

|Ki ∪ Kj| − |Ki ∩ Kj| ≤ 2kDyz

(3.9)

where Dyz represents the number of pairwise differences between haplotypes y and z and the

left side represents the number of k-mers exclusive to either i or j. The reason for the using a “≤”

in Equation 3.9 is to account for three possibilities: (1) if two SNPs are less than k bases apart, there

72

will be fewer than 4k k-mers not shared between i and j, (2) it’s possible for a SNP to turn one

k-mer into a different k-mer that’s already present elsewhere in a genome and (3) at higher ploidy

levels, if a segregating site is heterozygous in i and j then i and j will share all k-mers between

them.

To generalize beyond haploid organisms, we will replace the conversion factor of 2 in equation

3.9 above with a general function a(x):

|Ki ∪ Kj| − |Ki ∩ Kj| ≤ a(x)

∑

y<z

Dyz

(3.10)

where a(x) is a conversion factor that turns a number of pairwise differences into a number of

sample-exclusive k-mers as a function of x and

∑

y<z Dyz is all of the pairwise differences for any

pair of haplotypes y and z in i and j. The values that maximize a(x), maintaining the validity of

using a “≤” sign in Equation 3.10, are as follows:





a(x) =

2k
x2 ,

if 1 ≤ x ≤ 3.

k
2x−1 ,

if x ≥ 4.

(3.11)

When the population is haploid (x = 1), Equation 3.10 reduces to Equation 3.9. When x =

2 or 3, a(x) is maximized when i and j are homozygous for different alleles, creating 2k sample-

exclusive k-mers for every 4 or 9 pairwise differences, respectively. However, when x ≥ 4, the

situation that maximizes a(x) is one where the SNP exists on only one haplotype in i or j, creating

k sample-exclusive k-mers for every 2x − 1 pairwise differences.

Now we will convert the right side of Equation 3.10 into nucleotide diversity (π). Summing

across all pairs of individuals gives:

∑

i<j

|Ki ∪ Kj| − |Ki ∩ Kj| ≤ a(x)

∑

∑

i<j

y<z

Dyz

(3.12)

Next, we convert the right-hand side into genome-wide average π by dividing both sides by the

73

number of pairwise haplotype comparisons (Korunes and Samuk, 2021), which is the number of

individuals sampled n times ploidy x, choose 2:

∑

i<j

|Ki ∪ Kj| − |Ki ∩ Kj|
)

(

nx
2

(∑

≤ a(x)

i<j

)

∑
(

y<z Dyz
)
nx
2

∑

i<j

|Ki ∪ Kj| − |Ki ∩ Kj|
)

(

nx
2

≤ a(x)π

(3.13)

(3.14)

Next, isolating π on one side gives:

∑

i<j

|Ki ∪ Kj| − |Ki ∩ Kj|
)
nx
2

a(x)

(

≤ π

(3.15)

The intuition behind this formula is that, in a world where our assumptions are met, the number

of k-mers that are not shared between a pair of samples is bounded by a multiple of π. The need to

scale π by a(x) to get a bound reflects the fact that a given SNP can be captured by multiple k-mers,

but the exact relationship between pairwise differences and sample-exclusive k-mers depends on

ploidy.

Finally, we can substitute π for the standard formula for the expected value of π in a neutrally

evolving population at equilibrium (Tajima, 1996):

[∑

E

i<j

|Ki ∪ Kj| − |Ki ∩ Kj|
)
nx
2

a(x)

(

]

≤ 2xNeµ
1 + 4

3 2xNeµ

(3.16)

where Ne is effective population size and µ is the mutation rate (probability of mutation per

base pair per generation). The denominator on the right hand side of Equation 3.16 ensures that

the expected value of π saturates as it approaches it’s theoretical maximum of 0.75 (Tajima, 1996).

Altogether, equations 3.15 and 3.16 provide simple bounds for the average number of k-mers that

differentiate a pair of samples in terms of Ne and µ.

74

Figure 3.2: Comparison between a typical SNP-calling workflow and a k-mer counting work-
flow. A typical k-mer-based analysis begins the same as a SNP-based analysis: sequencing reads
from sampled genomes followed by trimming and other quality control steps on the reads. The
crucial difference comes down to whether the remaining reads are aligned to a reference sequence
(discarding any unaligned reads) or are used for k-mer counting. Workflows may vary in terms of
attempting to map k-mers to specific loci or calling variants other than SNPs (denoted with “?”).
The end result for each workflow is typically a matrix where each row is a different genomic vari-
ant, each column is a sample, and the elements represent the variant states. For SNPs, the standard
variant call format notation is to use 0/0 to represent homozygous reference genotypes, 0/1 or 1/0 to
represent heterozygous genotypes, and 1/1 to represent homozygous alternate genotypes. In k-mer
based analyses the variant states are instead counts of how many k-mers of a particular type were
observed in each sample.

75

Identifying variation with k-mers

Many of the earlier studies using k-mers to identify variants de novo focus on microbes, where

(despite small genome sizes) high levels of diversity make alignment and choosing a reference

genome difficult (Gardner and Slezak, 2010). However, similar approaches have now been ex-

tended to other taxa. One class of approaches calls SNPs de novo simply by comparing k-mers

between samples and searching for pairs of k-mers that differ at their central basepair (Gardner and

Slezak, 2010; Gardner and Hall, 2013; Bedo et al., 2016; Li et al., 2022). Another, more popu-

lar, class of methods relies on de Bruijn graphs, which are graphs where k-mers are nodes and two

nodes are connected if they share k − 1 bases (Compeau et al., 2011). SNPs in a sample then appear

as “bubbles” in these graphs and calling SNPs amounts to searching for these bubbles (Iqbal et al.,

2012; Leggett et al., 2013; Younsi and MacLean, 2015; Uricaru et al., 2015; Standage et al., 2019;

Gauthier et al., 2020). Finally, other approaches first identify k-mers that are present in all samples

(i.e. “anchor” k-mers) and finds paths between anchor k-mers through either local alignment or

traversing de Bruijn graphs (Audano et al., 2018; Kaplinski et al., 2021; Aylward et al., 2023). The

main drawback to these approaches is that the relative positions of the resulting variant calls are

usually unknown without further analysis (see Figure 3.2).

To call variants with known relative positions, it is possible to combine the strengths of k-

mers with either pangenomes or databases of previously-identified variants. This is because unique

variations in pangenomes will often be tagged by multiple unique k-mers. For example, one can

compare k-mers with a reference set of SNPs (Shajii et al., 2016; Pajuste et al., 2017; Denti et al.,

2019; Shi et al., 2023; Chu et al., 2024) or insertions (Puurand et al., 2019) to quickly genotype

samples without alignment. Or if a reference pangenome assembly is available, it’s possible to use

k-mers to infer the path through the pangenome corresponding to a given sample genome (Iqbal

et al., 2012; Ebler et al., 2022; Grytten et al., 2022; Häntze and Horton, 2023). However, similar

to common alignment-based SNP calling practices, the called variants will be limited to variants

present in the pangenome reference.

If references are not already available, k-mers can be especially helpful for identifying and as-

76

sessing variation in species with little prior knowledge. For example, the popular tools genomescope

and smudgeplot use the distribution of k-mer counts in a sample (i.e. the k-mer frequency spec-

trum) to rapidly estimate important parameters, including genome size, heterozygosity, and ploidy

structure all from unassembled read sets (Vurture et al., 2017; Ranallo-Benavidez et al., 2020). k-

mer frequency spectra can also identify unwanted variation that is potentially due to contamination

(reviewed in Cornet and Baurain (2022)) or sequencing error. k-mers have the useful property that

sequencing errors mainly manifest as k-mers that occur just once or a few times in a sample of reads

(Kelley et al., 2010). This is because random sequencing errors are unlikely to generate the same

k-mer many times and (if k is long enough, see Box 1) usually produce k-mers not found in the tar-

get genome. Thus, excluding low-copy k-mers from an analysis can mitigate sequencing error and

one can examine the k-mer frequency spectrum to choose an appropriate k-mer count cutoff (Zhao

et al., 2018). Although k-mer frequency spectra can be usefully mined for genome parameters,

their exact relationship to key population genetic parameters, such as measures of differentiation

or diversity, remains underexplored.

Measuring variation with k-mers

After calling variants, quantifying levels of variation is a crucial step in many population ge-

netics workflows. Because counting k-mers tends to be faster than alignment, k-mers could be

especially helpful for rapid, initial assessments of diversity that complement or guide pangenome

analysis. The standard approach to measure diversity in a population is to align sample sequences

to a reference genome then calculate either (1) the average level of heterozygosity across sites or (2)

the number of variants segregating in the sample. These are represented by Nei’s (Equation 3.17;

Nei and Li (1979); Nei and Tajima (1981)) and Watterson’s (Equation 3.18; Watterson (1975))

estimators of diversity, respectively:

π = (1 −

∑

i

p2
i )(

n
n − 1

)

(3.17)

77

θw =

∑

S
a=n−1
a=1

1
a

(3.18)

where n is the number of sequences in the sample, pi is the frequency of the ith allele at a

locus, and S is the number of segregating sites at a locus. While θw is based on a discrete count

of variants (S), π is shaped by the allele frequencies of variants and will be higher if variants are

common than if variants are rare (note that π is more commonly rewritten in terms of the average

number of differences between sequences (Korunes and Samuk, 2021)).

Can analogous measures of variation be derived from k-mers? There are over 30 valid mea-

sures of genetic difference based on k-mer counts used in previous literature (Benoit et al., 2016;

Zielezinski et al., 2017; Luczak et al., 2019; Zielezinski et al., 2019). However, the three most com-

mon k-mer dissimilarity measures are arguably Jaccard dissimilarity (Equation 3.19; Ondov et al.

(2016)), Bray-Curtis dissimilarity (Equation 3.20; Dubinkina et al. (2016); Benoit et al. (2020)),

and cosine dissimilarity (Equation 3.21, Choi et al. (2019)):

J(Ki, Kj) = 1 − Ki ∩ Kj
Ki ∪ Kj

B(Ci, Cj) = 1 − 2

4k∑

b=1

min(mb(Ci), mb(Cj))
mb(Ci) + mb(Cj)

C(Ci, Cj) = 1 −

Ci · Cj
∥Ci∥ × ∥Cj∥

(3.19)

(3.20)

(3.21)

where k is the length of k-mers to be included in the comparison, Ki and Kj are the set of k-mers of

length k present in a set of reads i and j, Ci and Cj are vectors of k-mer counts in a set of reads i and

j, and mi is a function that returns the relative frequency of the bth k-mer in a set (i.e. standardized

∑

∑

such that

4k
b=1 mb(Ci) and

4k
b=1 mb(Cj) equal 1). These measures work similarly to the classical π

and θw measures: the numerators are a measure of the number of sites that vary between individuals,

78

while the denominators are a measure of sample size (here the number of k-mers rather than number

of haplotypes). The main difference, however, is that k-mer-based measures of genetic dissimilarity

are not directly interpretable in terms of mutations, like π and θw can be with the assumption that

each SNP represents one mutation (Haubold et al., 2011; Haubold and Pfaffelhuber, 2012). Any

given k-mer may represent the combined presence of multiple mutations (Voichek and Weigel,

2020; Blanca et al., 2022) and any mutation can generate multiple new k-mers. Although this

means that k-mer-based genetic dissimilarity measures are only proxies for the true mutational

distance between individuals, they still effectively resolve relationships between lineages compared

to alignment-based measures (VanWallendael and Alvarez, 2022).

While Equations 3.19, 3.20, 3.21 have all been successfully used to measure genetic dissimi-

larity between samples in past studies, the formulae highlight their benefits and drawbacks. First,

Jaccard dissimilarity, perhaps the most commonly used k-mer dissimilarity metric (Ondov et al.,

2016; Ruperao et al., 2023), requires only knowing k-mer presence/absence patterns in samples

instead of k-mer counts and thus can take less memory to calculate than other k-mer-based dissim-

ilarity measures. However, as a consequence Jaccard dissimilarity may not capture the effects of

copy number variation and does not account for variation in coverage between samples, which af-

fects whether a given k-mer is called as “present” in a sample (VanWallendael and Alvarez, 2022).

Approaches that measure k-mer counts, like Bray-Curtis dissimilarity (Equation 3.20) and cosine

dissimilarity (Equation 3.21), can better account for these influences, but may require more mem-

ory for storing counts (Liu et al., 2017; Choi et al., 2019). To alleviate this memory problem, many

approaches calculate approximate k-mer dissimilarity measures with a small subset of k-mers (On-

dov et al., 2016; Zhao, 2019; Benoit et al., 2020; Pellegrina et al., 2020). An alternative approach

would be to instead compress the k-mer counts into a smaller array, keeping information from more

k-mers while simultaneously alleviating memory burdens (Melsted and Pritchard, 2011), but such

approaches have not been used to calculate genetic dissimilarity before. The relationship between

k-mer-based dissimilarity measures and π is also rarely explored. While some studies have inves-

tigated the relationship between π and Jaccard dissimilarity (VanWallendael and Alvarez, 2022),

79

other k-mer dissimilarity measures are possible and no studies to our knowledge have compared

these approaches for populations of varying levels of diversity - a key determinant of whether

alignment-based genotype calls are accurate (Cornish and Guda, 2015; Bush et al., 2020). In the

following sections, we investigate the efficacy of compressed and uncompressed k-mer-based mea-

sures of variation at capturing the true pairwise diversity of simulated populations.

Testing the efficacy of k-mer measures of variation.

We used simulations to investigate the relationship between the true value of π and genetic

diversity measured from k-mer based approaches from simulated sequencing reads.

Simulations

We simulated a neutrally evolving 100kb segment of the Arabidopsis thaliana genome. We first

forward simulated 300 neutrally evolving populations with SLiM 3 (Haller and Messer, 2019).

Each population consisted of 100 individuals simulated for 1000 generations with a uniform re-

combination rate of 10−8. To vary the diversity across simulations we varied the mutation rate

between 10−6 and 2 ∗ 10−4. For each simulation, we tracked the ancestry through tree-sequence

recording which records the geneological history of all samples (Haller et al., 2019). From these

trees, we generated sequences based on the Arabidopsis thaliana genome (chromosome 1 at po-

sitions 4,185,001-4,285,000) (Kent, 2002). We took a sub-sample of 10 individuals from the tips

of the trees and randomly assigned nucleotides to each SNP in the sample using msprime version

1.2.0 and tskit version 0.5.6 (Baumdicker et al., 2022). From the sub-tree of the sampled genomes,

we recorded the exact number of true average pair-wise differences (πt) across the sample of 10

individuals (20 chromosomes) using msprime (Baumdicker et al., 2022).

Generating k-mers

To test the performance of the k-mer measures on unaligned reads, we simulated reads for each

genome using an Illumina read simulator, InSilicoSeq 2.0.0 (Gourlé et al., 2019). We varied the

read count to later investigate the effect of coverage as described below. We generated two sets of

reads for each individual at coverages of 10x and 30x. After simulating reads, we counted k-mers

within the reads using KMC3 (Deorowicz et al., 2015; Kokot et al., 2017). We generated k-mer

80

count vectors with k = 10, 20, 30, and 40 for each individual. We used a threshold-based approach

to adjust the k-mer vectors to reduce the effects of sequencing errors; any k-mer count below

the threshold value was set to zero for a particular sample before any dissimilarity calculations.

For a threshold of 5, only k-mers with counts of 5 or more were considered when calculating the

difference between two or more groups of k-mer counts. We present data with a threshold of 5,

but note that a threshold of 0 is qualitatively similar with dissimilarity scores being slightly higher

overall. This practice of filtering out low-coverage k-mers is analogous to the common practice of

filtering out SNPs below a given minor allele frequency threshold (Asif et al., 2021). After k-mer

counting, we calculated the genetic dissimilarity of populations with the Bray-Curtis (Equation

3.20) and cosine dissimilarity (Equation 3.21) measures.

The effect of k on k-mer similarity metrics

Figure 3.3: Effect of k on Bray-Curtis dissimilarity. Bray-Curtis dissimilarity calculated from
simulated reads with a coverage of 30 where each point represents a sample of 20 chromosomes.
Displayed are 10-mers (yellow), 20-mers (green), 30-mers (blue), and 40-mers (purple) each with
a linear regression line for πt between 0 and 0.025. The scores for populations with a πt less than
0.025 is shown to the right.

Figure 3.3 shows the effect of k on the Bray-Curtis score with 30x coverage. We observe

a plateau in the scores when diversity exceeds π ≈ 2.5%. This plateau occurs because, when

diversity is high, SNPs cause most of the k-mers to be different between two samples. This effect

81

is especially true with high k values as one variant sampled in only one read can appear in up to

2k k-mers if it is sampled away from the edges of a read. The elevated number of k-mers at high

diversity means that it is harder to interpret differences in dissimilarity measures between samples

above this threshold, but this problem can be mitigated by using a lower k value, such as k=10

when the expected diversity in a sample is high. If πt is expected to be below 0.025, larger k values

can have higher precision and capture more unique k-mers in individual samples, better estimating

true diversity.

While the data presented in Figure 3.3 were simulated with a coverage of 30x, Figure D1 shows

that a coverage of 10x results in qualitatively similar scores when k=30. While coverage affects

the precision of the measures, the overall trends and relative rankings between simulations remain

consistent.

Figure 3.4: Effect of bloom filter on cosine dissimilarity. Cosine dissimilarity (triangle) and
counting bloom filter scores (circle) from 10-mers of simulated reads with a coverage 30 compared
directly against the Bray-Curtis score of each simulated set of 10 individuals (points).

82

The Counting Bloom filter approach to comparing genomes

Bloom filters are a data structure that can compress a k-mer count vector into a smaller array

(Melsted and Pritchard, 2011). We can compute the dissimilarity of two compressed vectors using

less memory and with increased efficiency as fewer entries are compared. Counting bloom filters

(CBFs) are modifications of bloom filters (Fan et al., 2000), and are used in alternative k-mer count

methods and error correction in sequencing data (Melsted and Pritchard, 2011; Roy et al., 2014;

Shi et al., 2010). CBFs compress the k-mer vectors through several hash functions that map to a

smaller array. Counting bloom filters will increase the k-mer count in the position as opposed to

setting the count to one if a k-mer has been mapped there (see Figure 3.5). Because of the hash-

functions and collisions, false positives are possible; here a “false positive” means that two k-mers

of different sequences may map to the same locations in the vector (i.e. have a hash collision).

However, identical mapping of two different k-mers is not a concern for relative comparison of

diversity, since the same hash functions are used for each sample so collisions are consistent across

samples. Collisions will cause diversity to be underestimated, but we can reduce collisions by

increasing the number of hash-functions used in the process (Melsted and Pritchard, 2011).

A CBF gives a standardized way to compare across species/experiments if the vectors for k-

mer counts are set to the same size and the same hash functions are used to generate the vectors

across data sets. We can also perform the same cosine dissimilarity measure on the CBF vector

that we use directly on the counts of k-mers. The cosine dissimilarity measure on the raw k-mer

count vectors has the same R2 to the compressed CBF vectors (0.97) when comparing the scores

to πt. Figure 3.4 shows that the cosine dissimilarity using the counting bloom filter is qualitatively

similar to the cosine dissimilarity with the raw k-mer counts. This similarity and the high R2

mean that the predictability of πt is maintained while using the smaller, more manageable CBF

data structure. Therefore, we can reliably use the CBF data structure as opposed to the raw k-mer

counts to calculate the cosine dissimilarity of two samples.

The memory usage of a k-mers vector for a sample with 10x coverage and k = 30 under

our simulations is around 4.5 MB per sample. The memory usage is reduced to 0.02 MB when

83

compressed to a 10,000 element array of unsigned 16-bit integers. Therefore, storing and using

k-mer counts for measuring diversity scale much better under the CBF data structure.

Figure 3.5: A conceptual schematic of a Counting Bloom Filter. When the first 4-mer, “ACGC”,
is hashed, the hash function h1 outputs 4 and h2 outputs 8. Both positions in the CBF array are
incremented by 2 as that is the count of ‘ACGC’ in the given k-mer count vector.

The effect of array size on CBF measures of diversity

We found that the size of the CBF array scales the cosine dissimilarity score and keeps the rank

of scores very similar as the array size changes. Since rank is maintained, we can take advantage of

much smaller arrays without losing much information about the relative diversity scores between

samples. Smaller arrays allow for less memory usage, and faster computing of cosine dissimilarity.

Supplementary Figure D2 compares the scores of a CBF array of size 20 million and 10 thousand on

the same simulations. We observe that the smaller array adjusts the scores down while maintaining

the same rank of scores. For example, these results show that if a population has the lowest cosine

score with the 20 million length array, it remains among the lowest scores with a 10 thousand length

array. This result means that it may be possible to adjust the vector size such that the cosine dis-

similarity scores are similar in magnitude and scale to the true pairwise nucleotide diversity which

can be useful for prediction of the score without alignment, and can make interpretation simpler.

However, the viability of this approach still needs to be investigated across different species and

84

genomic contexts, it is likely there is not a ‘one size fits all’ array size that would scale cosine

dissimilarity to match π universally. Our simulations only consider SNPs, neutral evolution, and

the Arabidopsis thaliana genome. It is still left to determine the effects of other mutations types

such as insertions, deletions, and inversions as well as other evolution types. Therefore we should

avoid the potential pitfalls of interpreting a scaled value directly as π unless future work shows that

to be appropriate.

Explaining variation with k-mers

A common first step to understanding the evolutionary forces shaping variation in populations

involves quantifying differentiation between populations. Patterns of differentiation in SNP geno-

types are often summarized and plotted using dimension reduction techniques, such as principal

component analysis (PCA) (Novembre and Stephens, 2008). k-mer genotypes are also amenable

to PCA and recover the same differentiation patterns as SNP-based PCA (Liu et al., 2017; Murray

et al., 2017; Rahman et al., 2018; Ho et al., 2019; Hrytsenko et al., 2022). Applying dimensional

reduction to k-mers can even differentiate species (Rosen et al., 2008; Aflitos et al., 2015; Bernard

et al., 2016; Linard et al., 2019; Boddé et al., 2022), something that can be difficult to do with

SNPs without multiple sequence alignment. Interestingly, k-mers from repetitive sequences may

not always differentiate populations that are clearly differentiated in terms of their SNP genotypes

(Renny-Byfield and Baumgarten, 2020), suggesting that identifying the set of k-mers that best dif-

ferentiate populations could be an important avenue for future research. Furthermore, while we are

aware of one study in yeast that estimates admixture proportions from k-mers (Shi et al., 2024),

more investigation is needed to see if this approach works for other species.

Besides polymorphism between populations, k-mers are also useful for explaining polymor-

phism patterns across genomes in terms of specific evolutionary forces. For instance, the k-mer

profile of a given locus is often predictive of the local recombination rate (Liu et al., 2012; Haubold

et al., 2013; Haubold, 2014; Frenkel et al., 2016; Al Maruf and Shatabda, 2019) and the local mu-

tation rate (Aggarwala and Voight, 2016; Carlson et al., 2018; Bethune et al., 2022; Adams et al.,

2023; Beichman et al., 2023; Liu and Samee, 2023). k-mers can capture information about re-

85

combination and mutation because these processes often associate with specific functional DNA

motifs (Myers et al., 2008; Růžička et al., 2017). However, k-mers are also intrinsically sensi-

tive to sequence changes and while there are some existing methods to identify new mutations

(Nordström et al., 2013; Ho et al., 2019) and sites of recombination (Fletcher et al., 2021) solely

from k-mers, further development is needed. Being able to predict fine scale variation in mutation

and recombination rates solely from the k-mer profile of a reference sequence would be extremely

beneficial because these processes frequently confound scans for sites of selection (Huber et al.,

2016). However, this would require further study of the relationship between k-mers and muta-

tion/recombination rates across wider ranges of species.

k-mers also offer opportunities to explore patterns of selection. One general assumption in

k-mer literature is if a k-mer is shared across all individuals of a population or species then it is

potentially under selection to be conserved (Bernard et al., 2016; Aylward et al., 2023). In contrast,

a k-mer that is not present in any individual is possibly selected against (although this assumes

that k is sufficiently small such that the total possible k-mer space is small, as might be true when

analysing small amino acid-derived k-mers) (Georgakopoulos-Soares et al., 2021). However, it

is unlikely that selection is solely responsible for patterns of k-mer sharing across individuals or

species, so more rigorous approaches are needed. One potentially more rigorous approach would

be to investigate k-mers that differentiate populations as candidates for loci underlying local adap-

tation, similar to FST (Kane and Rieseberg, 2007). Although, we are not aware of any papers

that employ this specific approach, identifying group-specific k-mers is a very common and useful

practice, with one example being the assembly of sex-specific sequences (Akagi et al., 2014; Ou

et al., 2017; Liao et al., 2020; Neves et al., 2020; Mehrab et al., 2021; Wu et al., 2021; Behrens

et al., 2022; Fong et al., 2023; Lichilín et al., 2023). There are some metrics of k-mer differentiation

in published literature already (Rahman et al., 2018), but they do not have the same interpretation

as FST . An alternative approach would be to march through a deBruijn graph (Aylward et al.,

2023) to find strings of k-mers with selective sweep-like patterns, mainly low diversity, high link-

age disequilibrium between k-mers, an excess of rare k-mers (Alachiotis and Pavlidis, 2018), high

86

differentiation between populations (Zhong et al., 2022), and high haplotype homozygosity (Klass-

mann and Gautier, 2022) which could be determined based on k-mer copy number (Vurture et al.,

2017; Ranallo-Benavidez et al., 2020).

There are only a few model-oriented studies that compare the k-mer spectrum observed in a

genome to a neutral expectation, mostly in the context of detecting selection on transcription factor

binding motifs (Gerland and Hwa, 2002; Ke et al., 2008; Raijman et al., 2008; Yeang, 2010; Gyorgy,

2023). The basic idea behind these approaches is to first derive a neutral substitution model - which

describes the probability of one nucleotide being substituted for another in a neutrally evolving

sequence (Ke et al., 2008; Raijman et al., 2008) - then measure deviations from that neutral model

according to the presence/absence of particular k-mers in a genome. Although these models are

able to detect selection on k-mers that exist in multiple places across a genome, as is the case with

binding motifs, it is unclear whether they could be applied to k-mers that represent unique genomic

sequences.

Challenges in investigating k-mers

Despite the wide potential of k-mers to be useful for population genetic studies, there are three

important limitations to keep in mind for most k-mer investigations. First, interpreting the biology

of specific k-mers is often a challenge. One option is to take the candidate k-mers identified from

an analysis and either align the k-mers themselves, the reads containing said k-mers, or assembled

reads containing said k-mers to a reference genome (Voichek and Weigel, 2020) or a database of

known sequences and motifs. While this is effective at pinpointing concrete type of variants at

play in a system, it partially defeats the purpose of using k-mers in the first place because k-mers

of interest may not be present in a reference genome. Second, since each sequencing error can

generate up to k erroneous k-mers and general practice is to discard reads with many unique k-

mers to reduce the effect of error rates (Zimin et al., 2013), analysis of k-mers typically requires

datasets with high coverage (>10x) and low sequencing error rates. These criteria can potentially

exclude long-read datasets (Vurture et al., 2017) or reduced-representation datasets where genomes

are sequenced at lower coverage to cut costs. Approaches that apply k-mers in lower coverage or

87

higher error-rate datasets are needed. Third, k-mer-based analyses can frequently involve hundreds

of millions or billions of unique k-mers. Storing and processing this many k-mers at once is often

not feasible even on high performance computing systems. Two approaches to solving this issue

are to subset one’s k-mers (Ondov et al., 2016; Benoit et al., 2020; Zhao, 2019; Yi et al., 2021),

or, as we discuss here, compress vectors of k-mer counts into an array of smaller size (Melsted

and Pritchard, 2011). However, these solutions are usually developed for specific contexts, such as

measuring genomic dissimilarity, and may not be applicable to every population genetic analysis.

Future studies need to carefully consider common approaches to decreasing the disk burden of

k-mer-based analyses and should explore new potential solutions.

These challenges to studying k-mers highlight how k-mer based methods should be viewed as

complementary to, instead of better than, pangenomes in many situations. Among other benefits,

pangenomes have ready biological interpretations and provide greater information on linkage be-

tween sequences. However, the fact that k-mers can be studied with fewer computing resources

and simpler sequencing data sets makes them potentially useful for either preliminary pangenomic

analyses or situations where constructing many reference quality genomes is not yet feasible.

Conclusions

Current literature demonstrates that k-mers are useful for identifying, measuring, and explain-

ing variation within populations without performing alignment. Through our own simulations of

neutrally evolving populations, we find that k-mer dissimilarity reliably scales with nucleotide

diversity and k-mer matrices can be compressed with minimal loss of dissimilarity information.

However, further development is needed to make k-mers amenable to a wider array of popula-

tion genetic tasks, especially the identification of selected loci and population structure. Further

developing alignment-free approaches to population genetics tasks will ultimately help guide and

complement the analysis of pangenomes.

Future directions

• Can we develop criteria for choosing values of k that account for population-level variation

in genome size or genome content?

88

• Can we develop k-mer-based estimates of population differentiation that are more similar to

FST ?

• Can we identify putative selective sweeps using k-mers?

• Can we develop applications of k-mers that are amenable to lower-coverage or pooled se-

quencing samples that are common in population genetics?

• k-mers are useful for getting preliminary estimates of important genome parameters that are

useful for tuning pangenome assembly (Ranallo-Benavidez et al., 2020). What other types

of preliminary pangenome analyses might k-mers be useful for?

• Can k-mers, either in combination with or without a pangenome reference, be useful for

rapidly estimating allele frequencies and performing demographic inference?

Data availability

Supplemental figures can be found in Appendix D. Scripts for simulation and data analysis can

be found at: https://github.com/williarj/kmers2024.

89

CHAPTER 4: SUMMARY STATISTICS COMPARABLE TO CONVOLUTIONAL
NEURAL NETWORKS FOR INFERRING TIMES TO FIXATION4

Abstract

Times to fixation (tf ) are an essential ingredient for understanding neutral theory and positive

selection. However, studies of tf have largely subsided in recent years and few methods exist to

predict the time to fixation for a fixed beneficial allele. A key difficulty in this prediction problem

is disentangling tf from the age of a sweep (ta) because multiple different combinations of tf

and ta can result in similar polymorphism patterns around the site of the sweep. We test whether

convolutional neural networks (CNNs) can potentially perform better on this problem by simulating

approximately 250,000 selective sweeps across five different demographic scenarios. The CNNs

achieve comparable performance to approximate bayesian compuation methods used in previous

approaches. The CNNs are better able to disentangle tf from ta for populations that are growing,

but also give statistically worse predictions than ABC when trained on populations cycling between

two different sizes. Altogether this suggests that perhaps few undiscovered signals remain in single

timepoint unphased genotype data that can be used to disentangle tf and ta in singe population

demographic scenarios.

Introduction

Adaptation requires increasing the frequency of beneficial alleles in a population, but these fre-

quency changes take time. How long does it take for beneficial alleles to spread? This question

was at the top of mind of naturalists in the early 20th century as there was doubt over how rea-

sonable natural selection was as an evolutionary mechanism (see Charlesworth (2020) for a fuller

historical treatment). The thinking was that if beneficial alleles change frequency only very slowly,

then maybe natural selection is not important. However, early models outlined how selection can

rapidly change allele frequencies, though most beneficial alleles are lost by drift Fisher (1923); Hal-

dane (1927). Nonetheless the rate of new mutations is high enough such that new beneficial alleles

are continually introduced and some eventually fix (Zhao et al., 2013). As it became appreciated

that multiple fixations can build up between species, further investigations of fixation times were

4At the time of writing, this chapter is unpublished.

90

motivated by observations of molecular substitution rates (Kimura and Ohta, 1969) and trait degra-

dation as a result of relaxed selection (Kimura, 1980). The theories from this foundational work

have since been extended to understand how time to fixation is shaped by changing environments

(Cui and Yuan, 2018; Kaushik and Jain, 2021) and population structure (Greven et al., 2016).

However, despite this foundation of knowledge, studies of times to fixation have subsided in

recent years such that two different papers spaced 8 years apart both argue that times to fixation

need more attention from modern methods (Zhao et al., 2013; Charlesworth, 2020). It is much

more common for studies to estimate coalescence times (Y. C. Brandt et al., 2022), variant age

(Bisschop et al., 2021), or allele frequency trajectories of currently segregating alleles (Stern et al.,

2019). One reason for this focus is that coalescent times are foundational to important population

genetic structures like ancestral-recombination graphs (Lewanski et al., 2024). Another reason is

because tf , in a simple model of a single additive de novo beneficial mutation, is tightly correlated

to the selection coefficient. The mean time to fixation for a single additive beneficial mutation in

a constant sized population is 2ln(2cNe − 1)/s where c is ploidy level (Otto and Whitlock, 2013).

This equation is sensitive to s more than Ne because s is outside of the logarithm. Thus, under

this model one could study the selection coefficient and get essentially the same information as

would be provided by tf . However, the exact relationship between time to fixation and s is more

complicated when arbitrary dominance, mating system variation, and demographic changes are

considered (Glémin, 2012). Selection coefficients and coalescent times thus do not fully convey

the timescale of fixation by themselves outside of simple scenarios.

How can we estimate tf ? Assuming a single de novo additive mutation that is sampled imme-

diately after sweep completion, there are simple models describing how the dip in diversity around

a selected site varies with tf (Coop, 2020). There are more complex models requiring different sets

of assumptions (He et al., 2020; Bisschop et al., 2021); however, not all study systems and situa-

tions will appropriately fit the assumptions in these models. As a result, there are several methods

that forgo explicit models and instead simulate sweeps and use approximate bayesian computation

(ABC) to estimate the timing of selection (Przeworski, 2003; Ormond et al., 2016; Nakagome et al.,

91

2019). Machine learning algorithms can also be trained on simulations and have proven useful for

a variety of population genetics tasks (Kern and Schrider, 2018; Flagel et al., 2019; Torada et al.,

2019; Sanchez et al., 2021; Whitehouse and Schrider, 2023), but have not yet been used to infer tf .

Our goal is to use machine learning models that can infer tf and are amenable to unphased

genotype data. Our hypothesis is that machine learning models will outperform ABC models and

could pick up on new signatures to differentiate between old, fast and young, slow sweeps (Hahn

and Mishra, 2025).

Methods

Our entire analysis is available as a snakemake workflow, packaged here: https://github.com/

milesroberts-123/selection-demography-cnn. This workflow includes all of the scripts used to sim-

ulate data, fit models, and generate figures as well as configuration files specifying the exact soft-

ware versions used for each step.

Simulations

Our simulated selective sweeps are generated using SLiM (v4.0.1, Haller and Messer (2019)).

The parameters for each simulation were generally drawn from uniform or log-uniform distribu-

tions. The exact distributions used for each parameter are listed in Table 4.1. Each simulation

initializes a diploid population of size NA individuals (i.e.

the ancestral population size) and a

chromosome of size L bp, then burns-in for 10NA generations with mutation rate µ and recombi-

nation rate R. At κ generations post-burn-in, a sweep mutation is introduced at position L/2 bp (i.e.

the middle of the simulated region). The beneficial mutation initially has a selection coefficient of

0, but once the frequency of the mutation reaches a value f0 the selection coefficient is changed

to a value s and given a dominance coefficient of h. After the frequency of the mutation reaches

a value f1, the sweep mutation is switched back to being neutral. After burn-in, the population

either has a custom demography (specified as a vector of time steps and a corresponding vector of

population sizes), or a demography determined by the following logistic map:

Nt+1 = rNt

)

(

1 − Nt
K

+ Nt

92

(4.1)

where Nt gives the population size at generation t and N0 = NA. This equation has the follow-

ing special properties (Phatak and Rao, 1995):

• r = 0, gives a constantly sized population.

• 0 < r < 2 and Nt < K gives a population that grows.

• 0 < r < 2 and Nt > K gives a decaying population.

√

• 2 < r <

√

6 gives a population that cycles between two different values.

•

6 < r < 3 gives a population that chaotically changes size. In practice, since our popula-

tion size is a discrete number of individuals, the population can have a long cycle if happens

to return to its initial population size of NA.

From now on, we will refer to these respective demographies with the terms “constant”, “growth”,

“decay”, “cycling”, or “chaotic”, respectively. For growth or decay demographies, we restrict r

to be 0 < r < 0.5 to simulate gradual, rather than nearly instantaneous, population size changes.

Because forward simulations slow down for larger populations, we also restrict K to be within 1

% - 50 % of NA. For each simulation, we then track how long it takes for the beneficial mutation

to fix. After fixation, we allow the simulation to run for an additional τ generations before taking

a random sample of size n individuals from the population. If a sweep was lost from a population

due to drift, we restarted the simulation to its post-burn-in state and initialized to a new random

seed. If a simulation was restarted 1000 times and still did not result in a complete sweep, then we

omitted that simulation from downstream analyses.

At the end of each simulation, we recorded the tf s from the completed simulation and randomly

sample n individuals (n = 128 for all analyses). We then downsampled the set of completed simu-

lations to create a uniform distribution of log10(tf ). Our downsampling procedure involved taking

the initial log10(tf ) distribution, dividing it up into bins of length 0.1, collecting the bin heights,

and randomly sampling simulations from each bin (without replacement) until all bins were the

same height. Our goal with this procedure was to include simulations with fixation times spanning

approximately 50 - 20,000 generations while still retaining > 11000 simulations per demographic

scenario. After downsampling, we randomly partitioned the remaining simulations to either the

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training dataset (80 %), the validation dataset (10 %) or the testing dataset (10 %).

For our simulations, we kept the following parameters constant, but it is possible to configure

different values in our workflow:

• L = 100 Kb

• κ = 1

• n = 128

• f0 = 0

• f1 = 1

In other words, we simulated only complete hard sweeps on chromosomes of size 100 Kb

introduced 1 generation after burn-in and randomly sampled 128 individuals, which is a similar or

larger sample size than most other sweep-related CNNs (Flagel et al., 2019; Torada et al., 2019;

Whitehouse and Schrider, 2023).

Parameter
NA

s

h
µ
R
τ
r
K

constant
U(1000,
10000)
LU(
log10(1/NA),
0)
U(0,1)
LU(-8.5, -7.5)
LU(-9, -7)
LU(0, 4)
0
NA

growth
U(1000,
10000)
LU(
log10(1/NA),
0)
U(0,1)
LU(-8.5, -7.5)
LU(-9, -7)
LU(0, 4)
U(0,0.5)
NA×U(1.01,
2)

decay
U(1000,
10000)
LU(
log10(1/NA),
0)
U(0,1)
LU(-8.5, -7.5)
LU(-9, -7)
LU(0, 4)
U(0,0.5)
NA×U(0.5,
0.99)

cycling
U(1000,
10000)
LU(
log10(1/NA),
0)
U(0,1)
LU(-8.5, -7.5)
LU(-9, -7)
LU(0, 4)
6)
U(2,
NA×U(0.8,
1.2)

√

chaotic
U(1000,
10000)
LU(
log10(1/NA),
0)
U(0,1)
LU(-8.5, -7.5)
LU(-9, -7)
LU(0, 4)
√
6, 3)
U(
NA×U(0.8,
1.2)

Table 4.1: Sampling distributions for key simulation parameters for the five different demo-
graphic scenarios included in this study. U is the uniform distribution and LU is the log uniform
distribution. The first number for each distribution is the minimum and the second number is the
maximum.

Selective sweep statistics

We calculated a suite of selective sweep summary statistics that were compatible with unphased

genotype data and useful for selective sweep analysis (Kern and Schrider, 2018), including the num-

ber of segregating sites (S), nucleotide diversity (π), Watterson’s theta (θW ), Tajima’s D (Tajima,

94

1989), variance in Tajima’s D, number of unique genotypes (Kern and Schrider, 2018), unphased

versions of h1; h2; h12; h123; h2h1 (Kern and Schrider, 2018), variance; skew; and kurtosis in the

distribution of genotype mismatches (i.e. the gkl statistic from Kern and Schrider (2018)), aver-

age genotypic correlation between pairs of SNPs (Rogers and Huff’s R2,Rogers and Huff (2009)),

Kim’s ω (Kim and Stephan, 2002), and unphased Messer’s hscan (Schlamp et al., 2016).

Image-based sweep representation

We converted the VCF file output from each simulation into an image representation using a

custom R script with ggplot (Wickham, 2016), ggnewscale (Campitelli, 2022), and cowplot (Wilke,

2020). This image representation is similar to (Flagel et al., 2019) but with some modifications for

unphased genotype data. Each image was a gray-scale matrix with n rows corresponding to the

sampled individuals and l columns corresponding to the l SNPs closest to the selective sweep site.

We clustered the rows using according to their manhattan distance using the complete clustering

algorithm in stats R package’s hclust function. The color of each element in the matrix could be

either black (homozygous genotype 0/0), grey (heterozygous genotype 0/1), or white (homozygous

genotype 1/1). We allowed l to reach a maximum of 128 SNPs and padded each image with black

columns if there were fewer than 128 SNPs in the sampled simulated population. Because each

column of the image represents a SNP we also created a vector of SNP positions to go along with

each image. These SNP positions were min-maxed normalized such that the SNP in the left-most

column of an image was position 0 and the SNP at the right-most column of the image was position

1.

Convolutional Neural Networks

We used the image-based representations of genotypes in the regions of selective sweeps to train

a convolutional neural network (CNN). Our CNN architecture was similar to Flagel et al. (2019),

with two main branches: one to process a sweep image and another to process the vector of SNP

positions associated with the columns of the sweep image. The image processing branch had three

convolutional layers, each followed by a pooling and dropout layer. The final dropout layer was

flattened and fed into a dense layer. The branch for processing the position information began with

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an input layer with 128 neurons (one for each column in the image), followed by a dense layer and a

dropout layer. The final dense layer for both the image-processing and position-processing branch

were concatenated and fed into a final dense layer with dropout before being fed into a single output

neuron.

We performed 60 iterations of bayesian hyperparameter tuning implemented in keras (v2.11.0,

Chollet and others (2015)) on each CNN. We chose 60 iterations because a random parameter search

with 60 iterations will sample a model in the top 5 % of the performance range with probability 95

% (Bergstra and Bengio, 2012). Each iteration involved 2 epochs of model fitting to the training

simulations and the final performance of the iteration was measured as mean squared error on the

validation simulations. During tuning, each convolutional layer could have any multiple of 16

filters between 16 and 128 and any layer could have a dropout rate between 0 and 0.99. All dense

layers could have any multiple of 32 neurons between 32, and 512. The first convolutional layer

was fixed to have a kernal size of 7 and a stride of 2, while the second and third convolutional layers

had a kernal size of 3 and stride of 1. All activation functions were rectified linear unit (ReLU)

functions (Banerjee et al., 2019).

After hyperparameter tuning, we then trained a final CNN with the best performing hyperpa-

rameters. Each training epoch fit the CNN in batches of 32 images. We trained the CNN until

validation performance did not improve for 20 epochs. After 20 epochs of no improvements, we

reverted the CNN to the configuration with the best performance on the validation data. This early

stopping criterion is meant to avoid overfitting the model on the training data. Final model perfor-

mance was assessed on the testing data and we performed 100 iterations of Monte-Carlo sampling

on each testing image to an estimate the uncertainty for each prediction, which we quantified as

the standard deviation in predictions across the Monte-Carlo samples. The mean of the predictions

across all the Monte-Carlo samples was used as the point estiamte for the outcome variable (tf )

from the CNN.

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Dense Neural Networks

We also constructed dense neural networks (DNNs) that were trained on only summary statis-

tics. Our DNNs had a input layer of 17 neurons (one for each summary statistic) followed by

three dense layers with dropout. Similar to the training of CNNs, we did 60 iterations of hyperpar-

mater tuning using bayesian hyperparameter optimization. During tuning, each of the dense layers

could have a multiple of 8 neurons anywhere from 16 to 512 and a dropout rate anywhere from 0 to

0.99. Final model performance was assessed on the testing data and we performed 100 iterations of

Monte-Carlo sampling on each testing image. The mean of the predictions across the Monte-Carlo

samples was used as our point estimate for the outcome variable (tf ).

Approximate Bayesian Computation

We also performed ABC to estimate tf using the rabc package (v2.2.1, Csilléry et al. (2012))

in R (R Core Team, 2022). Just like tuning hyperparameters of CNNs, performing ABC requires

making several choices, including the choice of regression method, the tolerance level, and the

type of point estimate used (i.e mean, median, or mode of the posterior distribution). Thus, similar

to ML, it is typically desirable to try many hyperparameter choices and use the combination of

hyperparameters with the best performance for final predictions. ABC approaches are typically

tuned with a cross-validation procedure: randomly drop a simulation from the training set and

then use the remaining training set to make a prediction for the dropped-out example, repeating

many times Csilléry et al. (2012). However, since part of our hypothesis hinged on comparing the

performance of ABC to CNNs and DNNs, we tuned ABC similarly to CNNs and DNNs to promote

comparability.

For each set of simulations, we tried 60 different configurations of ABC (similar to doing 60

iterations of hyperparameter tuning for CNNs and DNNs), including 4 different methods (rejection,

ridge regression, local linear regression, or neural network), 5 different tolerance values (0.025,

0.05, 0.1, 0.15, or 0.2), and three different point estimates (mean, median, or mode). For each

model configuration, we predicted tf for each simulation in the validation set using the training set

as the baseline. We quantified the performance of the model as the Pearson correlation between

97

the model’s predictions and the true value. For the best performing ABC model, we then measured

final performance by making predictions for the testing dataset, using the training dataset as the

background.

Partial R2 calculations

Another aspect we were interested in was the benefit of adding more selective sweep statistics to

explain variation in tf + ta because many studies that scan for sweeps simply focus on a handful of

site frequency spectrum (SFS) statistics like π and Tajima’s D (Teshima et al., 2006). To investigate

this, we calculated partial R2 values for the following statistics: S, log10(π), log10(θW ), Tajima’s

D, Variance in Tajima’s D, and the number of unique genotypes. These partial R2 values can be

interpreted as the proportion of variation not explain by these statistics that becomes explained by

our remaining statistics.

The procedure to calculate partial R2 is to first fit a ordinary least squares linear model, which

we will call the full model:

log10(tf + ta) ∼ S + log10(π) + log10(θW ) + D + Var[D] + number of unique genotypes + h1 + h2

+ h12 + h123 + h2h1 + gkl variance + gkl skew + gkl kurt + Messer’s hscan + Rogers and Huff’s

R2 + log10(Kim’s ω)

Then, fit a second model which we will call the reduced model:

log10(tf + ta) ∼ S + log10(π) + log10(θW ) + D + Var[D] + number of unique genotypes

Given the full and reduced model, the partial R2 measuring the added predictive power of

including more than just the SFS statistics is:

R2

partial =

SSE(reduced) − SSE(f ull)
SSE(reduced)

(4.2)

where SSE is the sum of squared errors.

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Results

Differentiating old, fast sweeps from slow, young sweeps

We performed a total of 250,000 SLiM simulations across 5 different demographic scenarios

(50,000 simulations per scenario), more than 79% of which successfully produced hard sweeps

with valid values of all 17 selective sweep statistics. For each of these simulations, we recorded tf

and ta for the sweep (Figure 4.1A). Most of these statistics produced clear, monotonic correlations

with tf + ta (Figure 4.1B, Figure E4-E7). However, for all of these statistics, there were divergent

combinations of tf and ta that produced similar values of the selective sweep statistics. For exam-

ple, sweeps that were old (ta > 1000) and fast (tf < 100) generally had similar π values to sweeps

that were younger and slower (tf > 1000, Figure 4.1C).

Figure 4.1: Separately estimating tf and ta from summary statistics leads to unidentifiability.
(A) Definition of time to fixation and sweep age. Time to fixation is the time between when a
sweep first arises and when it fixes. Sweep age is the time between the sweep fixes and when the
population is sampled. Example allele frequency trajectory comes from simulating a beneficial
allele with selection coefficient 0.5, dominance 0.5, and initial frequency of 1/1000 in a Wright-
Fisher population of size 1000. (B) Nucleotide diversity in the sweep region linearly scales with
the total lifetime of the sweep: tf + ta. (C) Young, slow sweeps (high tf and low ta) have similar
nucleotide diversity as old, fast sweeps (low tf and high ta).

Partial R2 of site frequency spectra statistics

Across the selective sweep statistics, the site frequency spectra statistics were generally posi-

tively correlated with each other and negatively correlated with the haplotype frequency statistics

(Figure 4.2A). The two statistics related to linkage disequilibrium, Rogers and Huff’s R2 and Kim’s

ω, were generally not correlated with any of the other statistics, except for being slightly correlated

to each other (Figure 4.2A). Across the 5 different demographic models, the partial R2 (i.e. the

99

added predictive power) of including statistics beyond simple SFS statistics was significant, with

values > 0.2 (Figure 4.2B). The growth demographic model had slightly higher partial R2 (0.26)

than the other scenarios and the chaotic demographic model had the lowest (partial R2 = 0.20).

Figure 4.2: Including more than site frequency spectrum statistics explains more variation
in tf + ta. (A) Pairwise Spearman correlations between unphased selective sweep statistics in
(B) Partial-R-squared values for a model with S, log10(π),
the constant demography scenario.
log10(θW ), Tajima’s D, Tajima’s D variance, and number of diplotypes.
In other words, these
values describe the amount of additional variation in tf + ta that explained when more than just
these selective sweep statistics are added to the model.

Convolutional neural networks vs summary statistics for predicting tf

For each of the 5 demographic scenarios, we built (1) Approximate Bayesian Computation,

(2) dense neural network, and (3) convolutional neural network models to predict tf . The pearson

correlation between the true value of tf and the predicted value of tf for the best performing version

of each model was generally > 0.7 (Figure 4.3, 4.4, E1-E3). By this measure of performance, the

three types of models were not significantly different for most of the demographic scenarios tested

(e.g. 95 % confidence intervals for constant demography: CNN = [0.705, 0.750], DNN = [0.719,

0.762], ABC = [0.731, 0.773]; Figure 4.3) with the exception of the cycling demography where the

CNN performed worse (r = 0.656 (CNN) vs 0.728 (DNN) vs 0.74 (ABC), Figure E3). All three

types of models also generally produced less accurate predictions for sweeps with a short tf , but

100

ta > 1000, incorrectly predicting that these sweeps had a much longer tf (Figure 4.3). The one

exception was the growth demography scenario where the CNN and DNN appeared to confuse old,

fast and slow, young sweeps less often (Figure 4.4).

Figure 4.3: CNNs perform similarly to DNNs and ABC in a constantly sized population. Per-
formance of best (A) CNN, (B) DNN, and (C) ABC models at predicting tf under a constant de-
mographic scenario. For each model, we list Pearson correlation coefficients between predicted
and true values (r), the number of test set examples (n), the 95 % confidence intervals for r, and
the p-value for testing if r is different from 0 (p). The purple line is the 1-1 reference line where
predictions and truth are equal. The yellow line is a least squares regression line. All points are
colored according to the age of the selective sweep in generations (log10 transformed).

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Figure 4.4: CNNs and DNNs and ABC in a growing population. Performance of best (A) CNN,
(B) DNN, and (C) ABC models at predicting tf under a constant demographic scenario. For each
model, we list Pearson correlation coefficients between predicted and true values (r), the number
of test set examples (n), the 95 % confidence intervals for r, and the p-value for testing if r is
different from 0 (p). The purple line is the 1-1 reference line where predictions and truth are equal.
The yellow line is a least squares regression line. All points are colored according to the age of the
selective sweep in generations (log10 transformed).

Discussion

We simulated approximately 250,000 selective sweeps across a range of parameters and demo-

graphic scenarios, then used those simulations to train CNNs, DNNs, and ABC models to predict

tf . Our hypothesis was that CNNs could potentially better disentangle tf from ta and thus be bet-

ter at predicting tf . Our simulations reflected a core challenge in the problem of distinguishing

tf from ta: sweeps with high tf and low ta (i.e. slow, young sweeps) leave similar signatures as

sweeps with low tf and high ta (i.e. fast, old sweeps, Figure 4.1C). Each of the 17 selective sweep

statistics we calculated explained additional variation in tf +ta (Figure 4.2). In the end, our models

trained on summary statistics (DNN and ABC) performed similarly to CNNs, which were trained

on images of genotype matrices (Figure 4.3, 4.4, E1-E3).

Similar to previous studies on estimating coalescent times (Y. C. Brandt et al., 2022), we noticed

a general bias where models overestimate low values of tf and underestimate high values of tf

(Figure 4.3, 4.4, E1-E3). This is mostly likely due to a general identifiability problem where old,

fast sweeps leave similar signatures as young, slow sweeps (Figure 4.1, E4-E7). However, in the

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growth demography scenario CNNs and DNNs were apparently better able to disentangle tf and ta

(Figure 4.4A and 4.4B) compared to ABC (Figure 4.4 C). This is possibly because two important

signatures of sweeps, an excess of common alleles in site frequency spectra and excessive linkage,

persist for longer in populations with higher Ne (Przeworski, 2002). Thus, the signatures of a short

tf persist for longer and prevent (or at least delay) confusion of old, fast sweeps with slow, young

sweeps. The second case where ABC and CNNs diverged was the cyclic demography scenario,

where the correlation between true and predicted tf was lower for the CNN than the DNN and

ABC, even though the DNN and ABC maintained a similar correlation compared to the constant

demography scenario (Figure E2). This is potentially because repeated population bottlenecks have

several consequences for the dynamics of sweeps (Wilson et al., 2014). In our case, if population

bottlenecks are softening the sweeps, then it’s possible that the CNN has trouble making predictions

for this type of sweep compared to DNNs and ABC which are fed statistics specifically tuned for

soft sweeps (H2H1, H123, etc.).

Summary statistics were overall effective at predicting tf from unphased genotype data (Figure

4.3B-C, 4.4B-C). Our hypothesis was that CNNs could potentially identify new signatures besides

known summary statistics and thus better predict tf (Hahn and Mishra, 2025). However, the fact

that ABC based on summary statistics performs the same or slightly better than a CNN suggests

unphased genotype matrices do not contain signals for tf that are not already captured by sum-

mary statistics. Many CNNs that classify selective sweeps use summary statistics partly to ease

interpretability (Kern and Schrider, 2018; Lauterbur et al., 2023), but previous studies have also

show CNNs trained on genotype matrices largely learn to reproduce summary statistics (Cecil and

Sugden, 2023). It is still possible though that additional signatures of tf not already described by

summary statistics do arise in other demographies not explored in this study.

Two general directions forward for improving tf predictions are to tweak the prediction algo-

rithms or include additional sources of data. Inclusion of time series data (Whitehouse and Schrider,

2023) or phasing information could conceivably aid tf prediction, but these are limited to cases

where the researchers have ancient samples or an understanding of a recombination landscape.

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Another option would be to include data from more than one population (Yair et al., 2021). Popu-

lation differentiation provides information on the split time between populations and if a sweep is

shared across populations, that would suggest that the sweep happened before the population split

(Yair et al., 2021). The difficulty here is that one needs to know or derive a model of the popula-

tion split and admixture beforehand so that it can be properly simulated. A combination of both

including additional data types and tuning models will most likely be the fruitful path forward.

Data availability

Supplemental figures are in Appendix E. Our entire analysis is available as a snakemake work-

flow, packaged here: https://github.com/milesroberts-123/selection-demography-cnn.

104

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APPENDIX A: SUPPLEMENTAL FIGURES FOR CHAPTER 1

General data curation

Figure A1: Histogram of mapping rates for RNA-seq runs. Bin width is 5 %.

139

Figure A2: Histograms of other key variables for the genes included in this study.

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0.00.20.40.62.55.07.510.0log(average expression in TPM)DensityA0500010000150000102030Number of genes in orthogroup (family size)Number of orthogroupsB012012Tajima's DDensityC0510150.30.40.50.60.7GC contentDensityDFigure A3: Violin plot of treatment-specificity in gene expression for six tissue categories.

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0.250.500.751.00fruit OR flowerleafrootseedshoot systemwhole plantTissueTreatment−specificityFigure A4: Violin plot of tissue-specificity in gene expression by treatment type. Only the five
treatments shown had samples from all six tissue types.

142

0.000.250.500.751.00PECO:0001062 control exposurePECO:0007105 abscisic acid exposurePECO:0007162 continuous light exposurePECO:0007319 warm/hot air temperature exposurePECO:0007332 cold air temperature exposureTissue−specificity of expressionFigure A5: Comparing dN and dS values returned from codeml and the custom biopython
script used in this study. Line shows exact match between codeml and biopython methods. Genes
with saturating divergence (dS > 1) were excluded from partial correlation analyses.

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Partial correlations on overall dataset

Figure A6: Partial correlation analysis for root tissue. Partial correlations for (A) dN , (B) πN ,
(C) Tajima’s D, and (D) direction of selection (DoS) based on root data. Average expression ex-
cludes values < 5 TPM and was calculated using only root tissue samples. Tissue-specificity was
calculated using only control runs across all tissue categories. Treatment-specificity was calculated
using only root tissue runs. The number of genes included in each partial correlation analysis (n)
is listed at the top of each heatmap.

144

-0.40.00.4Partialspearmancorrelationcoefficient0.170.11-0.010.090.040.44-0.260.02-0.130.15-0.050.08-0.140.01-0.18-0.18-0.15-0.13-0.23-0.3-0.12dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 14112A0.540.12-0.070.1-0.030.48-0.270.13-0.170.1-0.03-0.02-0.110.01-0.15-0.120.07-0.15-0.29-0.29-0.14-0.010.080.060.070.010.020.08πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 24513B0.020.030.5-0.03-0.210.08-0.03-0.120-0.150-0.17-0.31-0.27-0.140.050.060.070.010.020.09Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 24515C0.030.060.45-0.09-0.170.13-0.06-0.140.01-0.18-0.05-0.16-0.26-0.27-0.13Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 14064DFigure A7: Partial correlation analysis for whole plant tissue. Partial correlations for (A) dN ,
(B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on whole plant data. Average
expression excludes values < 5 TPM and was calculated using only whole plant tissue samples.
Tissue-specificity was calculated using only control runs across all tissue categories. Treatment-
specificity was calculated using only whole plant tissue runs. The number of genes included in
each partial correlation analysis (n) is listed at the top of each heatmap.

145

-0.40.00.4Partialspearmancorrelationcoefficient0.160.0500.120.040.33-0.270.02-0.230.08-0.040.08-0.16-0.02-0.12-0.18-0.17-0.24-0.14-0.28-0.1dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 13151A0.530.08-0.040.13-0.040.39-0.260.14-0.260.07-0.02-0.02-0.140-0.11-0.110.06-0.25-0.19-0.27-0.13-0.020.080.060.060.050.010.09πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 22292B0.010.030.4-0.02-0.290.04-0.03-0.14-0.01-0.110-0.27-0.21-0.25-0.120.050.060.060.070.010.1Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 22294C0.020.060.34-0.06-0.260.05-0.05-0.16-0.02-0.12-0.04-0.25-0.17-0.25-0.11Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13104DFigure A8: Partial correlation analysis for shoot tissue. Partial correlations for (A) dN , (B) πN ,
(C) Tajima’s D, and (D) direction of selection (DoS) based on shoot data. Average expression ex-
cludes values < 5 TPM and was calculated using only shoot tissue samples. Tissue-specificity was
calculated using only control runs across all tissue categories. Treatment-specificity was calculated
using only shoot tissue runs. The number of genes included in each partial correlation analysis (n)
is listed at the top of each heatmap.

146

-0.40.00.4Partialspearmancorrelationcoefficient0.150.07-0.030.050.070.51-0.270.02-0.190.08-0.040.08-0.1-0.09-0.12-0.17-0.15-0.11-0.26-0.26-0.11dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 13500A0.530.08-0.060.070.010.55-0.260.14-0.230.08-0.02-0.02-0.09-0.06-0.11-0.10.07-0.11-0.31-0.23-0.13-0.030.080.030.090.040.020.11πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 22727B00.040.56-0.02-0.260.06-0.02-0.09-0.07-0.110.01-0.12-0.31-0.21-0.130.050.030.090.060.020.11Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 22729C0.030.010.51-0.07-0.210.06-0.05-0.11-0.08-0.12-0.05-0.12-0.29-0.22-0.13Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13455DFigure A9: Partial correlation analysis for seed tissue. Partial correlations for (A) dN , (B)
πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on seed data. Average expression
excludes values < 5 TPM and was calculated using only seed tissue samples. Tissue-specificity was
calculated using only control runs across all tissue categories. Treatment-specificity was calculated
using only seed tissue runs. The number of genes included in each partial correlation analysis (n)
is listed at the top of each heatmap.

147

-0.40.00.4Partialspearmancorrelationcoefficient0.160.15-0.020.030.050.57-0.12-0.010.09-0.13-0.020.07-0.05-0.1-0.19-0.16-0.17-0.1-0.17-0.35-0.14dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 14223A0.530.16-0.090.0700.63-0.140.040.08-0.14-0.02-0.03-0.04-0.07-0.19-0.110.05-0.11-0.2-0.35-0.1700.070.030.110.020.020.08πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 25237B00.040.65-0.040.06-0.15-0.03-0.04-0.08-0.190-0.13-0.21-0.34-0.170.050.030.120.020.020.08Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 25242C0.040.030.58-0.030.08-0.14-0.05-0.05-0.1-0.19-0.05-0.12-0.19-0.34-0.16Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 14172DFigure A10: Partial correlation analysis for fruit or flower tissue. Partial correlations for (A)
dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on flower and fruit data.
Average expression excludes values < 5 TPM and was calculated using only fruit and flower tissue
samples. Tissue-specificity was calculated using only control runs across all tissue categories.
Treatment-specificity was calculated using only fruit and flower tissue runs. The number of genes
included in each partial correlation analysis (n) is listed at the top of each heatmap.

148

-0.40.00.4Partialspearmancorrelationcoefficient0.150.12-0.010.070.030.33-0.27-0.01-0.10.1-0.040.07-0.150.01-0.14-0.2-0.18-0.24-0.04-0.29-0.09dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 13550A0.530.15-0.080.06-0.020.38-0.280.13-0.140.07-0.04-0.01-0.140.03-0.14-0.130.06-0.28-0.09-0.25-0.11-0.020.080.070.050.050.010.07πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 22908B0.010.010.39-0.05-0.20.05-0.03-0.140.03-0.14-0.01-0.31-0.1-0.22-0.110.050.070.050.060.010.08Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 22913C0.040.030.34-0.08-0.140.08-0.05-0.150.01-0.14-0.06-0.28-0.06-0.25-0.11Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13506DPartial correlations on data subset by tissue type after SVA

Figure A11: Partial correlation analysis of root tissue after SVA. Partial correlations for (A)
dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on root tissue data after
applying SVA. Data was further subset to include only treatment groups with data from more than
one study before applying SVA. The number of genes included in each partial correlation analysis
(n) is listed at the top of each heatmap.

149

-0.40.00.4Partialspearmancorrelationcoefficient0.170.090.04-0.280.010.1-0.070.08-0.07-0.18-0.23-0.16-0.27-0.32-0.12dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 13984A0.530.15-0.04-0.290.140.05-0.05-0.01-0.06-0.15-0.170.09-0.34-0.29-0.1400.070.090.010.020.07πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 24104B0.05-0.040-0.03-0.07-0.140-0.38-0.25-0.140.050.10.010.010.07Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 24107C0.05-0.090.08-0.06-0.07-0.17-0.07-0.31-0.27-0.13Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 13937DFigure A12: Partial correlation analysis of whole plant tissue after SVA. Partial correlations for
(A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on whole plant tissue
data after applying SVA. Data was further subset to include only treatment groups with data from
more than one study before applying SVA. The number of genes included in each partial correlation
analysis (n) is listed at the top of each heatmap.

150

-0.40.00.4Partialspearmancorrelationcoefficient0.150.10.06-0.290.020.01-0.040.08-0.09-0.09-0.19-0.16-0.28-0.24-0.07dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 12909A0.530.14-0.03-0.30.16-0.02-0.03-0.02-0.07-0.08-0.120.07-0.35-0.2-0.1-0.030.080.10.0500.09πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 21557B0.05-0.02-0.06-0.03-0.07-0.070-0.37-0.18-0.10.050.10.0600.09Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 21562C0.05-0.07-0.02-0.06-0.09-0.08-0.05-0.33-0.2-0.08Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 12865DFigure A13: Partial correlation analysis of shoot tissue after SVA. Partial correlations for (A)
dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on shoot tissue data after
applying SVA. Data was further subset to include only treatment groups with data from more than
one study before applying SVA. The number of genes included in each partial correlation analysis
(n) is listed at the top of each heatmap.

151

-0.40.00.4Partialspearmancorrelationcoefficient0.160.080.04-0.280.010.19-0.040.08-0.02-0.07-0.18-0.19-0.2-0.19-0.04dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 12343A0.530.1-0.01-0.30.160.15-0.04-0.02-0.01-0.07-0.120.06-0.24-0.14-0.07-0.040.090.090.0400.09πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 20318B0.04-0.020.13-0.04-0.01-0.06-0.01-0.26-0.11-0.070.050.090.0600.1Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 20321C0.05-0.060.17-0.05-0.02-0.07-0.03-0.23-0.14-0.05Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 12308DFigure A14: Partial correlation analysis of seed tissue after SVA.Partial correlations for (A)
dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on seed tissue data after
applying SVA. Data was further subset to include only treatment groups with data from more than
one study before applying SVA. The number of genes included in each partial correlation analysis
(n) is listed at the top of each heatmap.

152

-0.40.00.4Partialspearmancorrelationcoefficient0.170.08-0.04-0.240.010.51-0.090.070.09-0.24-0.29-0.170.18-0.48-0.15dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 13958A0.520.08-0.07-0.250.110.5-0.0800.04-0.21-0.270.120.11-0.39-0.160.010.07-0.01-0.09-0.02-0.03πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 24748B-0.03-0.040.5-0.030.04-0.19-0.030.09-0.35-0.140.05-0.01-0.1-0.03-0.03Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 24752C0.05-0.120.51-0.080.08-0.24-0.130.18-0.46-0.16Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 13907DFigure A15: Partial correlation analysis of fruit or flower tissue after SVA.Partial correlations
for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) based on flower and
fruit tissue data after applying SVA. Data was further subset to include only treatment groups with
data from more than one study before applying SVA. The number of genes included in each partial
correlation analysis (n) is listed at the top of each heatmap.

153

-0.40.00.4Partialspearmancorrelationcoefficient0.140.10.04-0.300.12-0.050.070.02-0.1-0.22-0.19-0.08-0.22-0.03dSTreatment specificityAverage expressionGClengthdNdSTreatment specificityAverage expressionGCn = 12331A0.520.1-0.04-0.320.160.06-0.06-0.010.04-0.09-0.160.07-0.15-0.16-0.04-0.040.070.040.04-0.020.05πSTreatment specificityAverage expressionGClengthFamily sizeπNπSTreatment specificityAverage expressionGClengthn = 20327B0.01-0.030.03-0.030.04-0.07-0.01-0.17-0.12-0.030.040.030.06-0.020.06Treatment specificityAverage expressionGClengthFamily sizeTajima's DTreatment specificityAverage expressionGClengthn = 20332C0.03-0.090.09-0.060.02-0.1-0.06-0.12-0.17-0.04Treatment specificityAverage expressionGClengthDoSTreatment specificityAverage expressionGCn = 12291DPCA before and after SVA on data subset by tissue type

Figure A16: Effect of surrogate variable analysis on structure of leaf tissue data. (A) Effect of
SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal component
analysis of leaf tissue data before SVA. (D) Principal component analysis of leaf tissue data after
SVA. Color in the PCA plots represent different experimental treatments - legend is Figure A22.

154

2.55.07.510.02.55.07.510.0log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVALeafA0.000.250.500.751.000.40.60.81.0Treatment−specificity before SVATreatment−specificity after SVALeafB−300−200−1000100−1000100200300PC1:  15.14 %PC2:  7.01 %Leaf − Before SVAC−200−1000100−1000100200300PC1:  18.46 %PC2:  7.48 %Leaf − After SVADFigure A17: Effect of surrogate variable analysis on structure of root tissue data. (A) Effect of
SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal component
analysis of root tissue data before SVA. (D) Principal component analysis of root tissue data after
SVA.Color in the PCA plots represent different experimental treatments - legend is Figure A23.

155

2.55.07.510.012.52.55.07.510.012.5log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVARootA0.250.500.751.000.40.60.81.0Treatment−specificity before SVATreatment−specificity after SVARootB−100−50050100−200−1000PC1:  12.05 %PC2:  5.62 %Root − Before SVAC−1000100−150−100−50050PC1:  11.55 %PC2:  4.72 %Root − After SVADFigure A18: Effect of surrogate variable analysis on structure of whole plant tissue data. (A)
Effect of SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal
component analysis of whole plant tissue data before SVA. (D) Principal component analysis of
whole plant tissue data after SVA.Color in the PCA plots represent different experimental treat-
ments - legend is Figure A24.

156

2.55.07.510.02.55.07.510.0log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVAWhole plantA0.000.250.500.751.000.40.60.81.0Treatment−specificity before SVATreatment−specificity after SVAWhole plantB−150−100−50050100−1000100200PC1:  19.95 %PC2:  5.99 %Whole plant − Before SVaC−150−100−50050100−200−1000100200300PC1:  18.74 %PC2:  4.91 %Whole plant − After SVADFigure A19: Effect of surrogate variable analysis on structure of shoot tissue data. (A) Effect of
SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal component
analysis of shoot tissue data before SVA. (D) Principal component analysis of shoot tissue data after
SVA. Color in the PCA plots represent different experimental treatments - legend is Figure A25.

157

2.55.07.5369log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVAShootA0.250.500.751.000.40.60.81.0Treatment−specificity before SVATreatment−specificity after SVAShootB−1000100200−200−1000100PC1:  26.51 %PC2:  7.34 %Shoot − Before SVAC−200−1000100−1000100200PC1:  22.86 %PC2:  6.65 %Shoot − After SVADFigure A20: Effect of surrogate variable analysis on structure of seed tissue data. (A) Effect of
SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal component
analysis of seed tissue data before SVA. (D) Principal component analysis of seed tissue data after
SVA. Color in the PCA plots represent different experimental treatments - legend is Figure A26.

158

2.55.07.510.012.52.55.07.510.0log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVASeedA0.000.250.500.751.000.250.500.751.00Treatment−specificity before SVATreatment−specificity after SVASeedB−800−600−400−20000100200300400PC1:  2.53 %PC2:  1.58 %Seed − Before SVAC−300−200−1000100200300−750−500−2500PC1:  1.91 %PC2:  1.2 %Seed − After SVADFigure A21: Effect of surrogate variable analysis on structure of fruit and flower tissue data.
(A) Effect of SVA on average expression. (B) Effect of SVA on treatment-specificity. (C) Principal
component analysis of fruit and flower tissue data before SVA. (D) Principal component analysis
of fruit and flower tissue data after SVA. Color in the PCA plots represent different experimental
treatments - legend is Figure A27.

159

2.55.07.510.0369log average expression (excluding TPM < 5) before SVAlog average expression (excluding TPM < 5) after SVAFlower or fruitA0.000.250.500.751.000.250.500.751.00Treatment−specificity before SVATreatment−specificity after SVAFlower or fruitB−200−1000100−200−1000100PC1:  23.87 %PC2:  8.02 %Flower − Before SVAC−400−2000200−1000100200300PC1:  17.14 %PC2:  6.29 %Flower − After SVADLegends - PCA before and after SVA on data subset by tissue type

Figure A22: Legend for PCA before and after SVA, leaf data.

160

TreatmentPECO:0001021 ozone exposurePECO:0001034 Pseudomonas syringae plant exposurePECO:0001062 control exposurePECO:0007048 sodium chloride exposurePECO:0007048 sodium chloride exposurePECO:0007319 warm/hot air temperature exposurePECO:0007075 high light intensity exposurePECO:0007075 high light intensity exposurePECO:0007319 warm/hot air temperature exposurePECO:0007105 abscisic acid exposurePECO:0007162 continuous light exposurePECO:0007162 continuous light exposurePECO:0007315 zero gravity exposurePECO:0007200 short day length exposurePECO:0007200 short day length exposureflagellin 22 exposurePECO:0007200 short day length exposurePECO:0001050 limited nitrogen exposurePECO:0007200 short day length exposurePECO:0007075 high light intensity exposurePECO:0007200 short day length exposurePECO:0007233 fungal plant exposureBotrytis cinereaPECO:0007200 short day length exposurePECO:0007246 bacterial plant exposurePseudomonas fluorescensPECO:0007200 short day length exposurePECO:0007246 bacterial plant exposurePseudomonas syringaePECO:0007203 far red light exposurePECO:0007246 bacterial plant exposureVibrio vulnificusPECO:0007270 continuous dark (no light) exposurePECO:0007319 warm/hot air temperature exposurePECO:0007332 cold air temperature exposurePECO:0007373 mechanical damage exposurePECO:0007404 drought environment exposureFigure A23: Legend for PCA before and after SVA, root data.

161

PECO:0001046 limited phosphate exposurePECO:0001050 limited nitrogen exposurePECO:0001062 control exposurePECO:0007162 continuous light exposurePECO:0007162 continuous light exposurePECO:0007315 zero gravity exposurePECO:0007165 growth hormone exposureACCPECO:0007200 short day length exposurePECO:0007200 short day length exposureflagellin 22 exposurePECO:0007200 short day length exposurelimited copper exposurePECO:0007200 short day length exposurePECO:0001046 limited phosphate exposurePECO:0007200 short day length exposurePECO:0001049 limited magnesium exposurePECO:0007200 short day length exposurePECO:0007397 phosphorus nutrient exposurePECO:0007242 iron nutrient exposurePECO:0007246 bacterial plant exposurePECO:0007284 nitrogen macronutrient exposurePECO:0007319 warm/hot air temperature exposurePECO:0007373 mechanical damage exposureFigure A24: Legend for PCA before and after SVA, whole plant data.

162

Treatmentflagellin 22 exposurePECO:0001036 chemical stress exposuretunicamycinPECO:0001038 osmotic stress exposuremannitolPECO:0001046 limited phosphate exposurePECO:0001062 control exposurePECO:0007048 sodium chloride exposurePECO:0007105 abscisic acid exposurePECO:0007162 continuous light exposurePECO:0007162 continuous light exposurePECO:0007001 UV−B light exposurePECO:0007162 continuous light exposurePECO:0007048 sodium chloride exposurePECO:0007162 continuous light exposurePECO:0007075 high light intensity exposurePECO:0007162 continuous light exposurePECO:0007189 chemical exposureDMSOPECO:0007162 continuous light exposurePECO:0007319 warm/hot air temperature exposurePECO:0007189 chemical exposureDMSOPECO:0007196 light exposurePECO:0007200 short day length exposurePECO:0007200 short day length exposureflagellin 22 exposurePECO:0007200 short day length exposurePECO:0007319 warm/hot air temperature exposurePECO:0007200 short day length exposurePECO:0007332 cold air temperature exposurePECO:0007207 red light exposurePECO:0007246 bacterial plant exposurePECO:0007397 phosphorus nutrient exposurePECO:0007270 continuous dark (no light) exposurePECO:0007319 warm/hot air temperature exposurePECO:0007332 cold air temperature exposurePECO:0007397 phosphorus nutrient exposurePECO:0007404 drought environment exposureFigure A25: Legend for PCA before and after SVA, shoot data.

163

TreatmentPECO:0001046 limited phosphate exposurePECO:0001050 limited nitrogen exposurePECO:0001062 control exposurePECO:0007075 high light intensity exposurePECO:0007162 continuous light exposurePECO:0007165 growth hormone exposureACCPECO:0007200 short day length exposurePECO:0007200 short day length exposurePECO:0007319 warm/hot air temperature exposurePECO:0007200 short day length exposurePECO:0007332 cold air temperature exposurePECO:0007203 far red light exposurePECO:0007270 continuous dark (no light) exposurePECO:0007284 nitrogen macronutrient exposurePECO:0007332 cold air temperature exposurePECO:0007397 phosphorus nutrient exposurePECO:0007407 methyl jasmonate exposureFigure A26: Legend for PCA before and after SVA, seed data.

164

TreatmentPECO:0001062 control exposurePECO:0007162 continuous light exposureFigure A27: Legend for PCA before and after SVA, flower data.

165

TreatmentPECO:0001062 control exposurePECO:0007162 continuous light exposurePECO:0007248 greenhouse studyPECO:0007319 warm/hot air temperature exposurePECO:0007332 cold air temperature exposurePartial correlations, balanced subset after SVA

Figure A28: Partial correlation analysis of leaf tissue on balanced data subset after SVA. Partial
correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) after applying
SVA. Average expression and tissue specificity are calculated using only leaf tissue data. The
number of genes included in each partial correlation analysis (n) is listed at the top of each heatmap.

166

-0.40.00.4Partialspearmancorrelationcoefficient0.160.0700.040.050.43-0.280.02-0.160.07-0.040.08-0.14-0.03-0.13-0.17-0.17-0.18-0.15-0.26-0.1dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 12917A0.530.09-0.040.07-0.010.46-0.260.15-0.20.06-0.03-0.02-0.11-0.01-0.13-0.10.06-0.19-0.21-0.23-0.13-0.030.080.040.090.050.020.11πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 21746B0.020.030.47-0.01-0.230.05-0.03-0.11-0.01-0.130.01-0.2-0.22-0.22-0.120.050.040.090.060.020.11Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 21752C0.0400.44-0.07-0.180.05-0.05-0.14-0.02-0.13-0.04-0.2-0.18-0.22-0.11Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 12876DFigure A29: Partial correlation analysis of root tissue on balanced data subset after SVA.
Partial correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) after
applying SVA. Average expression and tissue specificity are calculated using only root tissue data.
The number of genes included in each partial correlation analysis (n) is listed at the top of each
heatmap.

167

-0.40.00.4Partialspearmancorrelationcoefficient0.170.120.010.030.040.29-0.250.02-0.110.07-0.050.08-0.11-0.08-0.21-0.2-0.15-0.19-0.23-0.36-0.16dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 13795A0.530.14-0.060.04-0.010.32-0.260.12-0.150.05-0.04-0.02-0.09-0.06-0.18-0.140.07-0.21-0.28-0.33-0.18-0.020.080.080.050.010.020.08πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 23515B0.030.010.33-0.04-0.20.04-0.03-0.1-0.06-0.17-0.01-0.24-0.28-0.31-0.170.050.080.050.010.020.08Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 23517C0.050.010.29-0.09-0.140.06-0.06-0.12-0.08-0.21-0.06-0.22-0.25-0.33-0.17Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13749DFigure A30: Partial correlation analysis of whole plant tissue on balanced data subset after
SVA. Partial correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS)
after applying SVA. Average expression and tissue specificity are calculated using only whole plant
tissue data. The number of genes included in each partial correlation analysis (n) is listed at the top
of each heatmap.

168

-0.40.00.4Partialspearmancorrelationcoefficient0.160.070.010.020.030.22-0.270.01-0.180.07-0.030.08-0.160.02-0.11-0.18-0.18-0.25-0.1-0.26-0.08dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 12608A0.530.11-0.040.04-0.010.27-0.260.13-0.220.06-0.03-0.02-0.140.03-0.1-0.110.05-0.28-0.14-0.24-0.11-0.040.080.070.040.0600.09πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 20944B0.030.020.28-0.02-0.260.05-0.03-0.140.03-0.10-0.29-0.14-0.22-0.110.050.070.040.0700.1Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 20947C0.040.010.22-0.06-0.210.06-0.05-0.160.02-0.11-0.04-0.28-0.12-0.22-0.1Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 12566DFigure A31: Partial correlation analysis of shoot tissue on balanced data subset after SVA.
Partial correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) after
applying SVA. Average expression and tissue specificity are calculated using only shoot tissue
data. The number of genes included in each partial correlation analysis (n) is listed at the top of
each heatmap.

169

-0.40.00.4Partialspearmancorrelationcoefficient0.160.09-0.01-0.020.060.39-0.270.01-0.160.07-0.040.08-0.11-0.1-0.11-0.18-0.16-0.16-0.26-0.24-0.11dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 12906A0.520.12-0.050.010.020.43-0.260.14-0.20.07-0.03-0.02-0.09-0.08-0.1-0.10.06-0.16-0.28-0.2-0.12-0.040.080.050.080.050.010.1πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 21405B0.020.020.44-0.02-0.240.07-0.02-0.09-0.08-0.090.01-0.17-0.28-0.18-0.120.050.040.090.060.010.11Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 21411C0.04-0.010.39-0.07-0.190.07-0.05-0.11-0.1-0.11-0.04-0.18-0.27-0.2-0.12Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 12866DFigure A32: Partial correlation analysis of seed tissue on balanced data subset after SVA.
Partial correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS) after
applying SVA. Average expression and tissue specificity are calculated using only seed tissue data.
The number of genes included in each partial correlation analysis (n) is listed at the top of each
heatmap.

170

-0.40.00.4Partialspearmancorrelationcoefficient0.160.170-0.050.020.53-0.14-0.010.04-0.16-0.030.07-0.1-0.04-0.2-0.18-0.17-0.14-0.15-0.37-0.13dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 14028A0.520.18-0.090.010.010.55-0.150.040-0.13-0.03-0.02-0.08-0.03-0.19-0.130.06-0.19-0.14-0.35-0.1600.070.060.08-0.010.010.06πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 24617B0.020.020.56-0.04-0.02-0.13-0.03-0.09-0.03-0.18-0.01-0.22-0.15-0.34-0.160.050.060.08-0.010.010.06Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 24621C0.0600.53-0.050.02-0.16-0.05-0.1-0.04-0.2-0.06-0.18-0.15-0.36-0.15Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13978DFigure A33: Partial correlation analysis of fruit or flower tissue on balanced data subset after
SVA. Partial correlations for (A) dN , (B) πN , (C) Tajima’s D, and (D) direction of selection (DoS)
after applying SVA. Average expression and tissue specificity are calculated using only fruit and
flower tissue data. The number of genes included in each partial correlation analysis (n) is listed at
the top of each heatmap.

171

-0.40.00.4Partialspearmancorrelationcoefficient0.150.110.010.1-0.010.24-0.290.01-0.060.1-0.050.07-0.140.02-0.14-0.21-0.17-0.260.03-0.3-0.09dSTissue specificityTreatment specificityAverage expressionGClengthdNdSTissue specificityTreatment specificityAverage expressionGCn = 13331A0.530.15-0.070.07-0.040.28-0.290.15-0.110.07-0.05-0.01-0.130.04-0.14-0.130.07-0.31-0.03-0.25-0.1-0.030.080.080.030.050.010.07πSTissue specificityTreatment specificityAverage expressionGClengthFamily sizeπNπSTissue specificityTreatment specificityAverage expressionGClengthn = 22346B0.0300.3-0.04-0.160.05-0.03-0.140.04-0.130-0.34-0.04-0.22-0.10.040.080.030.0600.08Tissue specificityTreatment specificityAverage expressionGClengthFamily sizeTajima's DTissue specificityTreatment specificityAverage expressionGClengthn = 22350C0.030.040.25-0.09-0.090.08-0.06-0.140.02-0.14-0.06-0.310.01-0.25-0.11Tissue specificityTreatment specificityAverage expressionGClengthDoSTissue specificityTreatment specificityAverage expressionGCn = 13287DEffect of omitting low-expression values on treatment-specificity vs expression level
correlation

Figure A34: Treatment specificity vs average expression correlation in root tissue. Correlation
between average expression in transcripts per million (TPM) and treatment specificity of genes
when low expression values (< 5 TPM) are included (A) vs excluded (B). Expression level and
treatment specificity were calculated using only data from root tissue samples. Line is a smoothing
line with 95 % confidence intervals and values in parentheses give spearman correlation.

172

−10−505100.40.60.81.0treatment specificity of expressionlog(average expression in TPM)root (−0.51)A2.55.07.510.012.50.40.60.81.0treatment specificity of expressionlog(average expression in TPM)root (0.1)BFigure A35: Treatment specificity vs average expression correlation in whole plant tissue.
Correlation between average expression in transcripts per million (TPM) and treatment specificity
of genes when low expression values (< 5 TPM) are included (A) vs excluded (B). Expression
level and treatment specificity were calculated using only data from whole plant tissue samples.
Line is a smoothing line with 95 % confidence intervals and values in parentheses give spearman
correlation.

173

−10−505100.40.60.81.0treatment specificity of expressionlog(average expression in TPM)whole plant (−0.56)A2.55.07.510.00.40.60.81.0treatment specificity of expressionlog(average expression in TPM)whole plant (−0.01)BFigure A36: Treatment specificity vs average expression correlation in shoot tissue. Correla-
tion between average expression in transcripts per million (TPM) and treatment specificity of genes
when low expression values (< 5 TPM) are included (A) vs excluded (B). Expression level and
treatment specificity were calculated using only data from shoot tissue samples. Line is a smooth-
ing line with 95 % confidence intervals and values in parentheses give spearman correlation.

174

−10−505100.40.60.81.0treatment specificity of expressionlog(average expression in TPM)shoot (−0.62)A3690.40.60.81.0treatment specificity of expressionlog(average expression in TPM)shoot (−0.01)BFigure A37: Treatment specificity vs average expression correlation in seed tissue. Correlation
between average expression in transcripts per million (TPM) and treatment specificity of genes
when low expression values (< 5 TPM) are included (A) vs excluded (B). Expression level and
treatment specificity were calculated using only data from seed tissue samples. Line is a smoothing
line with 95 % confidence intervals and values in parentheses give spearman correlation.

175

−10−505100.250.500.751.00treatment specificity of expressionlog(average expression in TPM)seed (−0.53)A2.55.07.510.00.250.500.751.00treatment specificity of expressionlog(average expression in TPM)seed (0.01)BFigure A38: Treatment specificity vs average expression correlation in fruit or flower tissue.
Correlation between average expression in transcripts per million (TPM) and treatment specificity
of genes when low expression values (< 5 TPM) are included (A) vs excluded (B). Expression
level and treatment specificity were calculated using only data from fruit and flower tissue samples.
Line is a smoothing line with 95 % confidence intervals and values in parentheses give spearman
correlation.

176

−10−505100.250.500.751.00treatment specificity of expressionlog(average expression in TPM)fruit and flower (−0.39)A3690.250.500.751.00treatment specificity of expressionlog(average expression in TPM)flower (0.03)BSimulating level-specificity correlation

Figure A39: Violin plot of correlations between expression specificity and expression level
across 1000 simulated expression matrices. In the two different simulations shown here, expres-
sion level was quantified either as average expression across all samples or average expression
across samples with expression values > 0. Each expression matrix was simulated by sampling
from a zero-inflated negative binomial distribution.

177

−0.4−0.20.0mean expressionacross all samplesmean expressionin samples where geneis expressedspecificity−expression level correlationAPPENDIX B: SUPPLEMENTAL METHODS FOR CHAPTER 2

The records that we used to estimate range size from GBIF occurrence data could not have any

issue codes except for the following:

• “AMBIGUOUS_COLLECTION”

• “AMBIGUOUS_INSTITUTION”

• “COLLECTION_MATCH_FUZZY”

• “COLLECTION_MATCH_NONE”

• “CONTINENT_DERIVED_FROM_COORDINATES”

• “COORDINATE_ROUNDED”

• “COUNTRY_DERIVED_FROM_COORDINATES”

• “COUNTRY_MISMATCH”

• “DEPTH_MIN_MAX_SWAPPED”

• “DEPTH_NON_NUMERIC”

• “DEPTH_NOT_METRIC”

• “DEPTH_UNLIKELY”

• “DIFFERENT_OWNER_INSTITUTION”

• “ELEVATION_MIN_MAX_SWAPPED”

• “ELEVATION_NON_NUMERIC”

• “ELEVATION_NOT_METRIC”

• “ELEVATION_UNLIKELY”

• “GEODETIC_DATUM_ASSUMED_WGS84”

• “INSTITUTION_COLLECTION_MISMATCH”

• “INSTITUTION_MATCH_FUZZY”

• “INSTITUTION_MATCH_NONE”

• “OCCURRENCE_STATUS_INFERRED_FROM_BASIS_OF_RECORD”

• “OCCURRENCE_STATUS_INFERRED_FROM_INDIVIDUAL_COUNT”

178

APPENDIX C: SUPPLEMENTAL FIGURES FOR CHAPTER 2

Exploring relationships in data before outlier removal

Figure C1: Tests of independence between life-history traits included in this study. Values
across the top of each plot give the results of a χ2 test of independence between each pair of life-
history traits.

179

Figure C2: Phylogenetic tree of species included in this study plotted alongside technical se-
quencing variables. The phylogenetic tree scale is in millions of years. We limited the number of
individuals sequenced per species to no more than 1200. CV stands for coefficient of variation and
bp for base pairs.

180

Figure C3: Correlations between technical sequencing variables and diversity. Each point is a
species. Values across the top of each plot give the Spearman’s correlation coefficient and p-value
(testing whether the correlation differs from zero) for each pairwise relationship. Each pairwise plot
includes one of three measures of diversity (Nucleotide diversity: A-C, Jaccard Dissimilarity: D-F,
Bray-Curtis dissimilarity: G-I) and one of three technical sequencing variables (mean coverage: A,
D, G; Coefficient of variation in bp sequenced: B, E, H; Number of individuals sequenced: C, F, I).
Purple line is a loess smoothing line with 95% confidence intervals shaded in gray. All logarithms
are base 10. Three species with nucleotide diversity values of 0 are omitted from plots involving
nucleotide diversity.

181

Figure C4: Effect of coverage on relationship between π and Jaccard dissimilarity. (A) shows
the relationship between k-mer diversity (Jaccard dissimilarity) and nucleotide diversity without
omitting species with ≤ 0.5x coverage or ≤ 1000 SNP calls. (B) shows the same relationship,
except these species with low coverage or low numbers of SNP calls are omitted. Each data point
is a species. All species’ points are colored by the log of average genome-wide coverage per in-
dividual (base 10) for that species. Purple lines are loess smoothing curves with 95% confidence
intervals shaded in gray. Values across the top of each plot are Spearman correlation coefficients
(ρ) and p-values that test whether each correlation coefficient differs from zero. Three species with
nucleotide diversity values of 0 are omitted from these plots.

182

Figure C5: Relationship between k-mer diversity, nucleotide diversity and variation in se-
quencing coverage. Each point is a species. Values across the top of each plot give the Spearman’s
correlation coefficient and p-value (testing whether the correlation differs from zero) for each pair-
wise relationship. Purple line is a loess smoothing line with 95% confidence intervals shaded in
gray. Points are colored by the coefficient of variation (mean/standard deviation) in bp sequenced
for each species. Three species with nucleotide diversity values of 0 are omitted from the plots.

183

Figure C6: Relationship between k-mer diversity, nucleotide diversity, and number of indi-
viduals sequenced. Each point is a species. Values across the top of each plot give the Spearman’s
correlation coefficient and p-value (testing whether correlation differs from zero) for each pairwise
relationship. Purple line is a loess smoothing line with 95% confidence intervals shaded in gray.
Points are colored by the logarithm of the number of individuals sequenced within each species
(base 10). Three species with nucleotide diversity values of 0 are omitted from the plots.

184

Figure C7: Independent methods of range size estimation correlate with each other. (A) Com-
pares the size of total size of native and invaded ranges from GBIF occurrence data and WCVP
range maps. (B) Compares native range estimates only (invaded ranges excluded) from GBIF oc-
currence data and WCVP range maps. (C) and (D) compare the full and native range estimates from
WCVP range maps and GBIF occurrence data, respectively.Every data point is a species. The solid
black lines are 1:1 reference lines where the different measures of range size are equal. The purple
lines are linear regression lines with 95 % confidence intervals in grey shading. Values across the
top of each plot give the Spearman’s correlation coefficient (ρ) and p-value testing whether corre-
lation differs from zero. One p-value is reported as zero because it was < 2.2 × 10−16, which is
the limit of precision for doubles in R. All logarithms are base 10.

185

Figure C8: Relationship between Bray-Curtis and Jaccard dissimilarity. Each point is a
species. Values across the top of each plot give the Spearman’s correlation coefficient and p-value
(testing whether correlation differs from zero). The p-value is reported as zero because it was
< 2.2 × 10−16, which is the limit of precision for doubles in R. Line is a linear regression with 95%
confidence intervals shaded in gray.

186

Figure C9: Relationship between the range size-squared height ratio and diversity without
correcting for evolutionary history, genome size, life cycle habit, mating system, or culti-
vation status. Each point is a species. Values across the top of each plot give the Spearman’s
correlation coefficient and p-value (testing whether correlation differs from zero) for each pair-
wise relationship. Each pairwise relationship involves the native range size-squared height ratio,
where native range size was estimated from GBIF occurrences (A, C, E), or WCVP range maps
(B, D, F), and one of three diversity measures (Nucleotide diversity: A-B, Jaccard Dissimiarity:
C-D, Bray-Curtis dissimilarity: E-F). Purple lines are loess smoothing lines with 95% confidence
intervals shaded in gray. Three species with nucleotide diversity values of 0 are omitted from plots
involving nucleotide diversity. All logarithms are base 10.

187

Population size proxy vs diversity relationships after controlling for phylogeny and
life-history variables, but not controlling for genome size

Figure C10: Partial phylogenetic regression between WCVP population size proxy and diver-
sity. Each point is a species and only species with > 0.5x mean coverage and > 1000 variant
sites were included in the regression. The regressions are organized according to whether invaded
ranges were excluded (A-C) or included (D-F) in the range size-squared height ratio. Range size
was estimated from WCVP range maps. Lines give the relationship between the pairs of plotted
variables after controlling for mating system, life cycle habit, cultivation status, and evolutionary
history. Before fitting the line, each response variable was scaled to a standard normal distribution
(mean = 0, variance = 1), then multiplied by the inverse of the Cholesky decomposition of the phy-
logenetic variance-covariance matrix to correct for phylogenetic relationships. The values at the
top of each plot give the slope of the partial regression ± one standard error and p-values testing
whether the slopes differ from zero. Dotted lines show the partial regression slope ± one standard
error. All logarithms are base 10.

188

Figure C11: Partial phylogenetic regression between GBIF population size proxy and diver-
sity. Each point is a species and only species with > 0.5x mean coverage and > 1000 variant
sites were included in the regression. The regressions are organized according to whether invaded
ranges were excluded (A-C) or included (D-F) in the range size-squared height ratio. Range size
was estimated from GBIF occurrence data. Lines give the relationship between the pairs of plotted
variables after controlling for mating system, life cycle habit, cultivation status, and evolutionary
history. Before fitting the line, each response variable was scaled to a standard normal distribution
(mean = 0, variance = 1), then multiplied by the inverse of the Cholesky decomposition of the phy-
logenetic variance-covariance matrix to correct for phylogenetic relationships. The values at the
top of each plot give the slope of the partial regression ± one standard error and p-values testing
whether the slopes differ from zero. Dotted lines show the partial regression slope ± one standard
error. All logarithms are base 10.

189

Figure C12: Partial phylogenetic regression between WCVP range size and diversity. Each
point is a species and only species with > 0.5x mean coverage and > 1000 variant sites were
included in the regression. The regressions are organized according to whether invaded ranges
were excluded (A-C) or included (D-F) in the range size estimates. Range size was estimated
from WCVP range maps. Lines give the relationship between the pairs of plotted variables after
controlling for mating system, life cycle habit, cultivation status, and evolutionary history. Before
fitting the line, each response variable was scaled to a standard normal distribution (mean = 0,
variance = 1), then multiplied by the inverse of the Cholesky decomposition of the phylogenetic
variance-covariance matrix to correct for phylogenetic relationships. The values at the top of each
plot give the slope of the partial regression ± one standard error and p-values testing whether the
slopes differ from zero. Dotted lines show the partial regression slope ± one standard error. All
logarithms are base 10.

190

Figure C13: Partial phylogenetic regression between GBIF range size and diversity. Each
point is a species and only species with > 0.5x mean coverage and > 1000 variant sites were
included in the regression. The regressions are organized according to whether invaded ranges
were excluded (A-C) or included (D-F) in the range size estimates. Range size was estimated
from GBIF occurrence data. Lines give the relationship between the pairs of plotted variables after
controlling for mating system, life cycle habit, cultivation status, and evolutionary history. Before
fitting the line, each response variable was scaled to a standard normal distribution (mean = 0,
variance = 1), then multiplied by the inverse of the Cholesky decomposition of the phylogenetic
variance-covariance matrix to correct for phylogenetic relationships. The values at the top of each
plot give the slope of the partial regression ± one standard error and p-values testing whether the
slopes differ from zero. Dotted lines show the partial regression slope ± one standard error. All
logarithms are base 10.

191

Figure C14: Partial phylogenetic regression between height and diversity. Each point is a
species and only species with > 0.5x mean coverage and > 1000 variant sites were included in the
regression. Lines give the relationship between the pairs of plotted variables after controlling for
mating system, life cycle habit, cultivation status, and evolutionary history. The statistics across the
top of each plot give the value of the slope of the lines (± the standard error), the p-value testing
whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. All logarithms are base 10.

192

Population size proxy vs diversity relationships after controlling for phylogeny, life-history
variables, and genome size

Figure C15: Partial phylogenetic regression between WCVP full range size-squared height
ratio and Jaccard dissimilarity, controlling for genome size. Each point is a species and only
species with > 0.5x mean coverage and > 1000 variant sites were included in the regression.
WCVP full ratio gives the ratio of range size to squared plant height, where range size includes in-
vaded ranges and is estimated from WCVP range maps. The partial regression controls for genome
size, mating system, life cycle habit, cultivation status, and evolutionary history. Before fitting the
line, each response variable was scaled to a standard normal distribution (mean = 0, variance =
1), then multiplied by the inverse of the Cholesky decomposition of the phylogenetic variance-
covariance matrix to correct for phylogenetic relationships. The values at the top of the plot give
the slope of the partial regression ± one standard error and p-values testing whether the slope differ
from zero. Dotted lines show the partial regression slope ± one standard error. All logarithms are
base 10.

193

Figure C16: Partial phylogenetic regression between WCVP native range size-squared height
ratio and diversity, controlling for genome size. Each point is a species and only species with
> 0.5x mean coverage and > 1000 variant sites were included in the regression. WCVP native ratio
gives the ratio of range size to squared plant height, where range size excludes invaded ranges and is
estimated from WCVP range maps. The partial regression controls for genome size, mating system,
life cycle habit, cultivation status, and evolutionary history. Before fitting the line, each response
variable was scaled to a standard normal distribution (mean = 0, variance = 1), then multiplied
by the inverse of the Cholesky decomposition of the phylogenetic variance-covariance matrix to
correct for phylogenetic relationships. The values at the top of the plot give the slope of the partial
regression ± one standard error and p-values testing whether the slope differ from zero. Dotted
lines show the partial regression slope ± one standard error. All logarithms are base 10.

194

Figure C17: Partial phylogenetic regression between GBIF range size-squared height ratio
and diversity, controlling for genome size. Each point is a species and only species with > 0.5x
mean coverage and > 1000 variant sites were included in the regression. The regressions are
organized according to whether invaded ranges were excluded (A-C) or included (D-F) in the range
size-squared height ratio. Lines give the relationship between the pairs of plotted variables after
controlling for mating system, life cycle habit, cultivation status, genome size, and evolutionary
history. The statistics across the top of each plot give the value of the slope of the lines (± the
standard error) and the p-value testing whether the slope differs from zero. Before fitting the line,
each response variable was scaled to a standard normal distribution (mean = 0, variance = 1), then
multiplied by the inverse of the Cholesky decomposition of the phylogenetic variance-covariance
matrix to correct for phylogenetic relationships. Dotted lines show the partial regression slope ±
one standard error. All logarithms are base 10.

195

Figure C18: Partial phylogenetic regression between WCVP range size and diversity, control-
ling for genome size. Each point is a species and only species with > 0.5x mean coverage and
> 1000 variant sites were included in the regression. The regressions are organized according to
whether invaded ranges were excluded (A-C) or included (D-F) in the range size estimates. Lines
give the relationship between the pairs of plotted variables after controlling for mating system, life
cycle habit, cultivation status, genome size, and evolutionary history. The statistics across the top
of each plot give the value of the slope of the lines (± the standard error) and the p-value test-
ing whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. Dotted lines show the partial regression slope ± one standard error. All logarithms
are base 10.

196

Figure C19: Partial phylogenetic regression between GBIF range size and diversity, control-
ling for genome size. Each point is a species and only species with > 0.5x mean coverage and
> 1000 variant sites were included in the regression. The regressions are organized according to
whether invaded ranges were excluded (A-C) or included (D-F) in the range size estimates. Lines
give the relationship between the pairs of plotted variables after controlling for mating system, life
cycle habit, cultivation status, genome size, and evolutionary history. The statistics across the top
of each plot give the value of the slope of the lines (± the standard error) and the p-value test-
ing whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. Dotted lines show the partial regression slope ± one standard error. All logarithms
are base 10.

197

Figure C20: Partial phylogenetic regression between height and diversity, controlling for
genome size. Each point is a species and only species with > 0.5x mean coverage and > 1000
variant sites were included in the regression. Lines give the relationship between the pairs of plot-
ted variables after controlling for mating system, life cycle habit, cultivation status, genome size,
and evolutionary history. The statistics across the top of each plot give the value of the slope of
the lines (± the standard error) and the p-value testing whether the slope differs from zero. Be-
fore fitting the line, each response variable was scaled to a standard normal distribution (mean =
0, variance = 1), then multiplied by the inverse of the Cholesky decomposition of the phylogenetic
variance-covariance matrix to correct for phylogenetic relationships. Dotted lines show the partial
regression slope ± one standard error. All logarithms are base 10.

198

Genome size vs diversity relationships after controlling for population size proxies,
phylogeny, and life-history variables

Figure C21: Partial phylogenetic regression between Jaccard dissimilarity and genome size,
controlling for WCVP full range size-height ratio. Each point is a species and only species with
> 0.5x mean coverage and > 1000 variant sites were included in the regression. WCVP full ratio
gives the ratio of range size to squared plant height, where range size includes invaded ranges and
is estimated from WCVP range maps. The partial regression controls for range size-squared height
ratio, mating system, life cycle habit, cultivation status, and evolutionary history. Before fitting
the line, each response variable was scaled to a standard normal distribution (mean = 0, variance
= 1), then multiplied by the inverse of the Cholesky decomposition of the phylogenetic variance-
covariance matrix to correct for phylogenetic relationships. The values at the top of the plot give
the slope of the partial regression ± one standard error and p-values testing whether the slope differ
from zero. Dotted lines show the partial regression slope ± one standard error. All logarithms are
base 10.

199

Figure C22: Partial phylogenetic regression between diversity and genome size, controlling
for WCVP native range size-height ratio. Each point is a species and only species with > 0.5x
mean coverage and > 1000 variant sites were included in the regression. WCVP native ratio gives
the ratio of range size to squared plant height, where range size excludes invaded ranges and is
estimated from WCVP range maps. The partial regression controls for genome size, mating system,
life cycle habit, cultivation status, and evolutionary history. Before fitting the line, each response
variable was scaled to a standard normal distribution (mean = 0, variance = 1), then multiplied
by the inverse of the Cholesky decomposition of the phylogenetic variance-covariance matrix to
correct for phylogenetic relationships. The values at the top of the plot give the slope of the partial
regression ± one standard error and p-values testing whether the slope differ from zero. Dotted
lines show the partial regression slope ± one standard error. All logarithms are base 10.

200

Figure C23: Partial phylogenetic regression between diversity and genome size, controlling for
GBIF range size-height ratio. Each point is a species and only species with > 0.5x mean coverage
and > 1000 variant sites were included in the regression. The regressions are organized according
to whether invaded ranges were excluded (A-C) or included (D-F) in the range size-squared height
ratio. Lines give the relationship between the pairs of plotted variables after controlling for mating
system, life cycle habit, cultivation status, range size-squared height ratio, and evolutionary history.
The statistics across the top of each plot give the value of the slope of the lines (± the standard error)
and the p-value testing whether the slope differs from zero. Before fitting the line, each response
variable was scaled to a standard normal distribution (mean = 0, variance = 1), then multiplied by the
inverse of the Cholesky decomposition of the phylogenetic variance-covariance matrix to correct
for phylogenetic relationships. Dotted lines show the partial regression slope ± one standard error.
All logarithms are base 10.

201

Figure C24: Partial phylogenetic regression between diversity and genome size, controlling
for WCVP range size. Each point is a species and only species with > 0.5x mean coverage and
> 1000 variant sites were included in the regression. The regressions are organized according to
whether invaded ranges were excluded (A-C) or included (D-F) in the range size estimates. Lines
give the relationship between the pairs of plotted variables after controlling for mating system, life
cycle habit, cultivation status, range size, and evolutionary history. The statistics across the top
of each plot give the value of the slope of the lines (± the standard error) and the p-value testing
whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. Dotted lines show the partial regression slope ± one standard error. All logarithms
are base 10.

202

Figure C25: Partial phylogenetic regression between diversity and genome size, controlling
for GBIF range size. Each point is a species and only species with > 0.5x mean coverage and
> 1000 variant sites were included in the regression. The regressions are organized according to
whether invaded ranges were excluded (A-C) or included (D-F) in the range size estimates. Lines
give the relationship between the pairs of plotted variables after controlling for mating system, life
cycle habit, cultivation status, range size, and evolutionary history. The statistics across the top
of each plot give the value of the slope of the lines (± the standard error) and the p-value testing
whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. Dotted lines show the partial regression slope ± one standard error. All logarithms
are base 10.

203

Figure C26: Partial phylogenetic regression between diversity and genome size, controlling
for height. Each point is a species and only species with > 0.5x mean coverage and > 1000
variant sites were included in the regression. The regressions are organized according to whether
invaded ranges were excluded (A-C) or included (D-F) in the range size-squared height ratio. Lines
give the relationship between the pairs of plotted variables after controlling for mating system, life
cycle habit, cultivation status, height, and evolutionary history. The statistics across the top of
each plot give the value of the slope of the lines (± the standard error) and the p-value testing
whether the slope differs from zero. Before fitting the line, each response variable was scaled
to a standard normal distribution (mean = 0, variance = 1), then multiplied by the inverse of the
Cholesky decomposition of the phylogenetic variance-covariance matrix to correct for phylogenetic
relationships. Dotted lines show the partial regression slope ± one standard error. All logarithms
are base 10.

204

APPENDIX D: SUPPLEMENTAL FIGURES FOR CHAPTER 3

The effect of coverage on k-mer measures

We observe in Figure D1 a complex interaction between coverage and the resulting Bray-Curtis

score depending on the length of k. With 30-mers, there is very little difference in the Bray-Curtis

score. However, 10-mers are more affected by changes in coverage. The overall trend and ranks

of the scores remain, but the Bray-Curtis score is generally higher with 10x coverage.

Figure D1: Effect of k and coverage on Bray-Curtis dissimilarity. Bray-Curtis scores calculated
with 10-mers (left) and 30-mers (right). For each sample, reads with 10x coverage (gray/square) and
30x coverage (black/pentagon) were simulated. After simulating reads, the k-mers were counted
and then the Bray-Curtis score is calculated. Each point represents one sample with a specific k
and coverage.

205

Adjusting array size of Counting Bloom Filter

Figure D2: Effect of bloom filter size on cosine dissimilarity. A comparison of cosine similarity
scores of CBFs with an array size of 20 million (left) and 10,000 (right). The black line shows the
1-to-1 mapping of πt to the dissimilarity score. Note the difference in the scales of the scores on
the y-axes, and that the right panel shows points that are closer to the 1-to-1 line.

206

APPENDIX E: SUPPLEMENTAL FIGURES FOR CHAPTER 4

Figure E1: Comparison of best (A) CNN, (B), DNN, and (C) ABC models at predicting times
to fixation for the decay demography scenario.

Figure E2: Comparison of best (A) CNN, (B), DNN, and (C) ABC models at predicting times
to fixation for the cycle demography scenario.

207

Figure E3: Comparison of best (A) CNN, (B), DNN, and (C) ABC models at predicting times
to fixation for the chaos demography scenario.

Figure E4: tf vs ta for statistics based on the site frequency spectrum for sweeps in constant-
sized populations.

208

Figure E5: tf vs ta for unphased H statistics and unphased Messer’s Hscan in constant-sized
populations.

Figure E6: tf vs ta for Kern’s gkl statistics in constant-sized populations.

209

Figure E7: tf vs ta for Rogers’ R2 and Kim’s ω.

210