Abstract
The process of forming new species is the driving force behind the diversity of life on Earth. Yet, we have not answered the basic question: why are species unevenly distributed across taxonomic groups and geographic settings? This is because we lack the means to directly compare aspects of population (lineage) divergence across unrelated species because taxon-specific effects make comparisons difficult or impossible. Here, we present a new solution to this challenge by identifying the information signature of diverging lineages, calculated using partial information decomposition (PID), under different evolutionary scenarios. We show in silico how the informational decomposition of genetic metrics varies over time since divergence. Calculating PID over 97,200 lattices reveals that the decomposed nodes of Tajima’s D, θW, and π have strong information signatures, while FST was least useful for discriminating among divergence scenarios. The presence or absence of gene flow during divergence was the most detectable signature; mutation rate and effective population size (Ne) were also detectable whereas differences in recombination rate were not. This work demonstrates that PID can reveal evolutionary patterns that are minimally detectable using the raw metrics themselves; it does so by leveraging the architecture of the genome and the partial redundancy of information contained in genetic metrics. Our results demonstrate for the first time how to directly compare characteristics of diverging populations even among distantly related species, providing a foundational tool for understanding the diversity of life across Earth.
INTRODUCTION
Many central themes of evolutionary biology deal with how new species differentiate from related ones, but many core questions about speciation remain unsolved. Among these are how often lineages diverge in isolation or with gene flow, how often certain geologic or geographic settings promote speciation, and which taxonomic groups are most affected (1). Answering these questions depends upon comparing aspects of lineage (population) divergence, such as the amount of genetic divergence and gene flow between populations, in different taxa. Yet, the organismal traits that cause population pairs of species to diverge differently under a common set of conditions also influence the measures of population diversity and divergence (e.g., π, FST) that we use to characterize those differences, making it difficult to compare characteristics of diverging populations across distantly related species (Figure 1). Some such aspects can be tackled using the site frequency spectrum (SFS; 2), but the SFS has limitations because different histories can produce similar geometric distributions (3).
(A) Organisms may be affected differently by the same extrinsic events (e.g., formation of a river). (B) Genic and intergenic parts of the genome are differentially shaped by extrinsic (e.g., barrier to gene flow) and intrinsic (e.g., selection) forces that can be quantified with genetic metrics (e.g., FST, Tajima’s D), the values of which will vary by taxon. (C) We calculated common metrics (π-nucleotide diversity, D-Tajima’s D) in genic and intergenic regions of the genome on simulated data and (D,E) calculated mutual information on these genetic statistics which is decomposed over a lattice into unique, redundant, synergistic information terms given by (F).
In this study we take an altogether new approach by assessing the information signal of genetic metrics under different evolutionary scenarios. Multivariate analysis is helpful when individual variables convey partial knowledge about a system of study. Taking an example from the kinetic theory of ideal gasses, macroscopic observables such as temperature and pressure provide information about the microscopic state of the system; they each provide some information, but it is only when taken together that the density of the gas can be determined solely through observation of those properties. We can characterize and identify observables which have utility for answering specific questions about the system under study by decomposing the information the observables provide over combinations of those observables, identifying the redundant, unique and synergistic contributions of each observable. Along this vein, we chose to decompose the multivariate information signal of different combinations of genetic metrics calculated on datasets simulated under a suite of evolutionary scenarios. Using the architecture of the genome, this information signal is effectively standardized within a dataset making comparison across species possible.
DNA is an information storage molecule because it encodes directions for the formation of proteins and those directions are heritable across generations. Information processing across levels of biological organization is also thought to be a fundamental, measurable characteristic that sets living systems apart from nonliving systems (4). It may therefore be appropriate that information theoretic measurements reveal higher-order patterns about biological evolution, which fundamentally based on the context-dependency and patterned cross-generational transmission of genetic information (DNA) (5).
Claude Shannon introduced mutual information in 1948 to quantify the degree to which observations of some random variable reduces the observer’s uncertainty about another variable (6). Mutual information and its related information-theoretic quantities have been applied in biology (7-10), machine learning and data science (11-13), complex systems theory (14-16), and ecohydrology (17). As with linear measures of correlation, mutual information can be generalized to a multivariate setting where total correlation (18), dual total correlation (19) and interaction information (20) are exemplars. We calculated the PID formalized by Williams and Beer (21) over eight common genetic metrics applied to ‘genic’ and ‘intergenic’ regions of genetic data simulated for diverging populations. PID decomposes the total correlation between a target variable and a collection of source variables over a lattice (Figure 1E). The nodes of the lattice represent redundant, unique and synergistic contributions of combinations of the source variables, and each combination yields a non-negative information contribution.
Application of PID to genomic data here (Figure 1) is based on the premise that how information about evolutionary processes is encoded across genomic elements (e.g., genes), and how that information decomposes over genetic metrics (e.g., FST), provides information about both the evolutionary and organismal context of that sequence (Figure 1B). Consider populations diverging without gene flow (i.e. reproduction). We expect these populations to accrue divergence in intergenic (noncoding) elements of the genome faster than coding elements, and this discrepancy would be higher in genetically isolated populations than if the populations had some level of gene flow (22). To understand the information signature of genetic metrics under different evolutionary settings, we simulated 60 kbp of sequence data for individuals in each of two populations under 24 evolutionary scenarios. Evolutionary scenarios varied mutation rate (μ), recombination rate (r), effective population size (Ne), and gene flow (M). We sampled genomes of 10 individuals per population at 10 timepoints per simulation to obtain a timeseries signature during divergence. Simulated genomes contained 10 ‘intergenic’ regions that evolved unconstrained and 10 ‘genic’ regions that evolved under purifying selection where mutations that changed the amino acid from the ancestral state penalized the individual’s fitness (see methods). We simulated 100 replicates per scenario which were concatenated and used to calculate eight genetic metrics (FST, Da, Tajima’s D, θW, π, S, Fu’s FS, and DXY) per genomic region. For each set of genetic metrics, we calculated PID over lattices constructed from combinations of up to four variables using genic/intergenic as the target variable. Results yielded 97,200 PID lattices that contained 7,648,800 nodes in total. The results were vectorized over timepoints to determine how the information of nodes within lattices varied with time since divergence and we grouped these into temporal patterns using k-means clustering. Results for the long time series (105 generations) showed that nodes within decomposed lattices exhibited partial information patterns that fell into seven timeseries classes (Figure 2B). Most nodes were invariable (N=302,207), while some reliably decreased (Class 2, N=196) or increased (Class 3, N=128). Complex time signatures are in SI (Figure S1).
(A) Change in information content (ΔΠ) in single-variable decompositions across evolutionary scenarios. A higher value reflects a greater ability for that variable to discriminate among evolutionary scenarios. (B) Time series summary of how nodes from the decomposed lattices change over time. Patterns were summarized into seven time-signature clusters using k-means. Three of those are shown here: flat (grey, N=303151), increasing (turquoise, N=368), and decreasing (orange, N=196).
We find that information signatures depended almost entirely on the informativeness of a few nodes representing key genetic metrics and not on the set of variables being decomposed (Figures 2a, 3, S2, S3). The only exception to this pattern was Tajima’s D, which increases in informativeness over time in the absence of gene flow (Figure 3c). When included among the variables, Tajima’s D redistributed information over the lattice, making other nodes more informative even though Tajima’s D itself was rarely (if ever) part of the node with an informative time signature (class 2 or 3) or an ability to discriminate among scenarios (e.g., Figure 3d). The variables lending greatest informativeness (i.e. ability to discriminate among evolutionary scenarios) were nucleotide diversity (π) and Watterson’s estimator (θW) (Tajima’s D was influential for the reason described). The types of nodes with greatest discrimination potential were either unique or redundant; synergistic nodes were never discriminatory among evolutionary scenarios (Figure 3). We interpret this to be due to the redundancy or nonindependence of the genetic metrics and not a behavior of PID itself.
(A) Partial information for Unique{Pi} from [DXY, FST, π] over 100,000 generations, which shows the strongest discrimination capability between gene flow (pink, green) and no gene flow (blue) following a critical threshold when the pattern stabalizes. (B) The 18-node decomposition lattice for [DXY, FST, π], from which a single node {π} is plotted in (a). (C) Single variable decomposition of Tajima’s D showing increasing partial information for {D} with time since divergence in the absence of gene flow, causing redistribution of information across lattices. (D) Comparison of AUC values for a receiver-operator curve (ROC) for the ability of π to discriminate between presence/absence of gene flow using gPID (green) versus raw π values (orange).
Our exploration of evolutionary parameter space (Ne, M, r, and μ) reveals the presence of gene flow to be easiest to detect (Figure 3), which is consistent with results based on analysis of the Site Frequency Spectrum (SFS, 23). The discrimination is binary rather than continuous, identifying the presence not the degree of gene flow in our simulations. However, sampling more rates of gene flow could further resolve this pattern. The variable most able to recover the gene flow signal is π while Tajima’s D and FST reflect the presence/absence of gene flow only at higher mutation rates (Figure S4). Mutation rate is recovered to a lesser extent than the presence of gene flow and is reflected best by Fu’s FS. Segregating sites (S) and Watterson’s estimator (θW) reflect differences in mutation rate but are also influenced by presence/absence of gene flow (Figure S5). S and θW yielded identical results (Figure S5); this is expected because θW is calculated by correcting S for the number of samples observed which was consistent across our simulations. This nonetheless demonstrates PID’s consistency to record equivalent patterns despite calculation over raw values of different scales. If we varied sample size, these metrics might no longer produce identical decompositions.
No nodes effectively discriminated recombination rate, which could be because our metrics did not reflect patterns of genomic linkage. We speculate that decomposing alternative genetic metrics could recover a difference in recombination rates. For differences in Ne (Figure S6), the magnitude of partial information was slightly inflated under smaller effective population sizes (Figure 3). It is possible that additional targeted exploration of ‘node-space’ with a focus on discriminating among Ne could uncover a more useful decomposition.
We assessed the reproducibility of PID results by selecting four scenarios (M = 0, μ = 2.5 × 10−7, Ne ∈ {500,1000}, r ∈ {0, 10−7}) and simulating 300 additional genetic datasets in batches of 100 per scenario (Table S2). As done previously, the eight metrics were calculated, their results concatenated per batch, and PID computed independently on the three new batches of simulations. To assess technical variance, lattices across the four batches (the one original batch and three additional replicate batches) were compared using mean-absolute-difference (MAD). Results showed the MAD between any pair of lattices never exceeded 1e-4 bits, which is orders of magnitude below the differences we observe between evolutionary scenarios. In other words, the technical variance among genetic simulations and genomic PIDs was exceedingly small.
After as few as 104 generations, many of the PID nodes exhibited partial information near the theoretical maximum of 1 bit. However, it is expected that before divergence begins that all nodes have negligible partial information because the populations start out from a single sequence. To confirm this, we re-simulated the first 104 generations and sampled at a higher frequency (every 103 generations) with the expectation that the nodes exhibiting non-zero partial information would rapidly depart from 0 bit. Results are consistent with the assumption that the information content begins at 0 bit; results show it becomes nonzero following the first generation. There is slightly less stability in the early generations evidenced by the greater number of time series clusters compared to the 105-generation dataset (N=20 vs N=7 groups, respectively; Figure S1 vs. S7). With additional work, it is possible that this latency period could be leveraged to measure aspects of very early divergence (≲ 1000 generations, Figure S8). Aside from slightly reduced power in early generations to discriminate among evolutionary scenarios, the information patterns stabilize once a critical threshold is met (Figure 3A). This reliability-of-pattern is favorable for the applicability of PID to genomic datasets of wild populations in which age-of-divergence dates can be reasonably presumed to be ≳3000 generations.
Several broadscale observations can be made from these results about both the behavior of information decomposition and its application to genetic metrics and evolution. First, it is surprising that FST, which is one of the most broadly used and interpretable statistics in evolution, was the least informative when decomposed, while more fundamental measures of diversity (π, S or θW) drove information patterns. Also, key variables determined information patterns rather than the combination of variables in the decomposition (the variable set). This finding is important because computing PID on a large number of variables is costly and difficult to interpret; it suggests that small lattices with a few key variables are sufficient for many purposes, and new applications of PID should assess patterns within small decompositions first. Likewise, we were surprised to find that the unique and redundant nodes far outweighed information in synergistic nodes. We interpret this to be a property of the genetic metrics analyzed, particularly their non-independence under different evolutionary settings. If, unlike other multivariate statistics, PID can leverage interpretational power from the nonindependence of variables, that may be its biggest strength. It is possible that synergistic terms could gain power under more evolutionarily complex scenarios, such as reduced gene flow from a physical barrier in combination with differential adaptation, which remains a stumbling block for analyses using the SFS (24).
A final, related discovery was that combinations of variables exhibited “additive” discriminatory behavior. As an example, the single variable decomposition [S] displayed sensitivity to both M (presence/absence) and μ (Figure 4a). The single variable decomposition of [D] reflected primarily presence or absence of gene flow (M, Figure 3c). The two-variable decomposition [S,D] presents a simple combination of [S] and [D], with a primary signal of gene flow/no gene flow (present in both [S] and [D]) and a secondary signal of μ (present in [S]). The discovery of additive behavior among variables should help biologists perform a targeted search of ‘decomposition-space’ for lattices that will discriminate between biological parameters of interest. In theory, choice of metrics to decompose can be optimized based on which properties of a diverging population are of interest.
(A) Information variability across evolutionary simulations for single decomposition of segregating sites (S) and (B) Tajima’s D. Higher values (yellow) indicate stronger discrimination ability. Grey bars at bottom and right are guides for the heatmap pattern expected for each variable simulated (μ - mutation rate, M - gene flow, Ne - effective population size, r - recombination rate). For example, Tajima’s D is discriminatory among gene flow whereas S detects both gene flow and mutation rate. (C) The information signatures are additive, wherein {S, Tajima’s D} displays the additive effects from panels A and B and the signal in heatmap (C) is exactly 1/4 the sum of the signals in (A) and (B), demonstrating true additivity.
In summary, answering many outstanding questions in speciation and macroevolution depends on our ability to compare characteristics of diverging lineages across diverse organisms. We outline a new information theoretic approach to do this which leverages the architecture of the genome and redundancy of well-known genetic metrics. Application of PID here shows the presence/absence of gene flow is easily detectable by several decompositions. But more importantly, PID is a new class of tool that can be applied to other variables and the large decomposition-space can be mined for nodes targeted to specific evolutionary characteristics and timepoints.
MATERIALS & METHODS
Genetic simulations & evolutionary analysis
We simulated population divergence scenarios in SLiM v3.3.2 (25) using house scripts that automated the parameter generation, simulations, and molecular statistic calculations with EggLib (Evolutionary Genetics and Genomics Library) v3.0.0b21 (26). The automated pipeline was written in python, Jinja, linux, and Eidos and allows for parallelizing SLiM simulations to be run in batch on high performance computing clusters (available at https://github.com/mmoral31/SLiM_pipeline); it processes intermediate files, and calculates diversity statistics. All simulations started with a random seed and initial genome sequence of 60,000 nucleotides divided into ten “coding” and ten “noncoding” regions of 3,000 nucleotides each to simulate a small eukaryotic genome. The “coding” and “noncoding” regions were composed of a random assortment of 1,000 codons (excluding stop codons) and 3,000 nucleotides, respectively. We simulated data under all combinations of the following four parameters: recombination rate r ∈ {0, 10−7}, mutation rate μ ∈ = {2.5 × 10−7,2.5 × 10−6}, symmetrical gene flow M ∈ = {0.0,0.01,0.10}, and effective population size Ne ∈ {500,1000}, for a total of 24 scenarios (Table S1). To simulate purifying selection on coding regions and calculate fitness, in SLiM we translated and compared the coding regions for each individual per generation to its initial (ancestral) amino acid sequence. The individual was assigned a reduced fitness of 0.9 based on one amino acid mismatch and 0.5 for two or more mismatches. Simulations were run for 100,000 generations and VCF (Variant Call Format) files were generated every 10,000 generations; each parameter combination was replicated 100 times, yielding 2,400 VCF files that were then concatenated for genetic calculations.
To calculate genetic metrics, we used EggLib (Evolutionary Genetics and Genomics Library) on the concatenated VCFs generated by SLiM. The house script calculated eight statistics: FST, Da, Tajima’s D, θW, π, S, Fu’s FS, and DXY in each 3-kb coding and noncoding region individually (i.e. window size and step size of 3 kb; Table S2). Whether the region was ‘coding’ or ‘noncoding’ was recorded in the output for each window. These statistics were then passed into Imogen.jl for PID calculations.
Technical variation & early divergence simulations
To assess technical variance of simulations and its effect on PID calculations, we ran an additional 300 replicates (for a total of N=400 per parameter combination) for a subset of parameters: no gene flow, mutation rate = 2.5 × 10−7, with both population sizes of 500 and 100, and with and without recombination (Table S2). To confirm the theoretical assumption that the genetic metrics hold no information at the immediate onset of divergence, and to test the reliability of decompositions during early-divergence, we ran another set of simulations focusing on early time-series sampling. For this we re-ran all 24 scenarios for 10,000 generations and sampled every 1,000 generations, with another 100 replicates per scenario (Table S2). The VCFs were generated at generation two instead of generation one due to software limitations. This set of data also consisted of 2,400 VCF files that were concatenated for genetic analysis.
Partial information decomposition
Implementation & running of analyses
The preceding analysis provided a sequence of values for each genetic metric along with whether or not the associated 3-kb region was a coding or noncoding region. In other words, over the length of the genome, we had a binary vector 
representing whether or not a given region is coding, and eight vectors 
– one for each evolutionary statistic estimated in the region. We treated these as sequences of observations of random variables G and di which allowed us to ask how the information about whether a given region is coding decomposes over combinations of the evolutionary statistics using the PID formalism. However, in general the 
are real-valued, and since the Williams and Beer PID formalism is limited to discrete-valued data, we performed a discretization process on each 
. For the sake of simplicity and to reduce systematic errors due to the short genome lengths (60,000 bp), we opted to use a “mean-threshold” binning scheme: the jth value of 
, written here as dji, was replaced with a 1 if it was greater than the average of all values in 
, and 0 otherwise. This yielded a set of eight binary vectors 
. PID could then be applied to the resulting vectors 
and 
In this process, the multivariate mutual information between 
and 
was estimated and subsequently decomposed into a sum of non-negative terms, roughly describing the degree to which combinations of the variables 
provided unique, redundant, and synergistic information about whether or not a given region is coding. Due to computational limitations of PID, the process can only be applied to at most five of the evolutionary statistics at time. In this work, we limited calculations to at most combinations of four statistics considered simultaneously to limit computation time and keep the amount of data generated manageable. We carried out the computation for all lattices with between one and four of the genetic metrics as source variables and the region classification 
as the target to yield 162 PID lattices per simulation per sampled generation for a total of 97,200 lattices containing 7,648,800 nodes in total.
Information clusters & informative metrics
For each node in each lattice per simulation, we created a vector of the partial information (Π) for that node over time. Many of the resulting curves had similar temporal dynamics but started out with a different initial value. To account for this, we shifted each vector by subtracting the zeroth element, element-wise, to make all vectors start at the same zero value. To reduce the 764,880 vectors into classes of information signatures over time, we performed k-means clustering based on Euclidean norm distance in the 10-dimensional space formed by the 10 sampled timepoints. We used the “elbow method” to identify a reasonable number of clusters for the k-means analysis. That is, we plotted the explained variance against the number of clusters for a range of values of k, and selected k at or near a prominent elbow (bend) in the curve. Based on this method, we chose a 7-means clustering to reduce the number of patterns on which to conduct further analyses without oversimplifying the diversity of results.
Limitations
We note several aspects of this analysis that could affect the results obtained. First, the genomes simulated were small (60 kbp) relative to the gigabasepair-size genome of vertebrates, for example. For simplicity, simulations did not incorporate standing variation in the population, which is a simplification of reality. Further, certain technical aspects of the Williams and Beer formalization of PID, such as the lack of localizability, continuity and differentiability, leave room for further exploration. More recent work by (27,28) made progress in addressing these points, and have arguably developed notions of information decomposition that are easier to interpret (which may be important for the future of PID for biological applications). The choice to use Williams and Beer’s formalism necessitated a binning procedure on the computed evolutionary quantities before analysis, and the particular method of binning can have effects on the resulting decompositions. It would be worth considering different discretization methods (i.e. 29-31) to assess the sensitivity of the results presented in this work.
DATA AVAILABILITY
Code is available on Github: https://github.com/elife-asu/gPID and https://github.com/mmoral31/SLiM_pipeline
All data files will be made available upon publication.
AUTHOR CONTRIBUTIONS
DGM and GAD conceived of this project; DGM, SIW, and GAD supervised analyses. MM carried out genetic simulations and pipeline automation, DGM performed information theoretic analyses. GAD and DGM drafted and all authors critically revised the manuscript.
FUNDING
This work was funded by NSF-EAR award #1925535 to GD, NSF GRFP to MM, award #61184 from the John Templeton Foundation to SIW.
ACKNOWLEDGEMENTS
We thank the Baja GeoGenomics consortium for useful feedback and discussions and Kenro Kusumi for support.
