queeems: Quantify the Extent of Evolutionary Evidence in Molecular Sequences

Introduction

queeems is a R1 package hosted on the Bioconductor2 platform. The package is useful for assessing the quality of genetic data that are meant for evolutionary investigations. Some invaluable methods and resources are available in the literature for similar purpose. These include: DAMBE3, SatuTe4, TreSpEx5, PAUP6, RASA7 and MUST8. Interested reader may consult Fleming et al9 for a short review of some of the remarkable contributions to the science of molecular saturation analyses. These existing techniques are designed primarily for investigations related to phylogeny reconstruction and many of them rely on adhoc frequentist-based hypotheses tests. queeems contains Bayesian functions that can be used to assess the quality of genetic data that are intended for use in phylogeny reconstruction or in broader evolutionary inquiries, including natural selection investigations. In this practical application of some of the primary functions in the package, I present analyses of simulated sequences to illustrate how to get the most out of this resource.

Installation

Use the following code to install queeems from the Bioconductor platform.

if(!requireNamespace("BiocManager", quietly=TRUE))
    install.packages("BiocManager")
BiocManager::install("queeems")

The developmental version of the package that contains the most recent updates, is available under the devel tab of the Bioconductor platform and can be installed by running the following code, just before the final install line above: BiocManager::install(version="devel").

Data

The molecular sequences analysed herein are generated with the scoup10 package that allows genetic sequences to be simulated following a simplified Ornstein-Uhlenbeck (OU) process. A R file that contains the simulation code and a folder that contains the simulated data are available in the inst/extdata/example/ folder within the queeems package. The folder also contains all the external data files relevant to any of the discussions presented here.

In general, genetic sequence alignments were simulated on a balanced bifurcating evolutionary tree that has 64 leaves. For each data simulated, the length of the internal and the terminal branches were all kept equal to the same value. Some of the other necessary scoup inputs that are common for all the simulated data analysed here, are presented in the table below.

Inputs of the scoup genetic sequence simulator that were kept unchanged for all the executions that generated the data sets analysed with queeems herein. Any other input of the simulator not listed in the table were left at their default values.
Number of terminal tree branches 64
Number of codon sites 100
Effective population size 1000
Variance of synonymous selection coefficients 0.01
OU reversion rate 0.00
OU asymptotic variance 0.00
Number of simulated data replicate per query 5

Example application of queeems

There are two primary uses of queeems: (a.) to generate molecular entropy estimates and (b.) to make inference about the saturation status of genetic data. I demonstrate both cases by considering two sources of molecular substitution saturation: (a.) time since divergence from common ancestor as measured by tree lengths11 and (b.) speed of molecular substitution as quantified by the magnitude of positive natural selection pressure12.

Tree length analyses

In the code chunk below, I used the queeems::seqSaturation function to obtain site-wise and overall Bayes factors so that I may assess the saturation level of the corresponding simulated sequences. I also used the queeems::molentropy function to obtain Shannon13 information entropy indices with respect to nucleotide, codon and amino acid bases. In addition to the tabulated scoup inputs, with respect to all the data sets generated for this section, the variance of non-synonymous selection coefficients (\(\sigma_{`n`}^{2}\)) was set equal to 0.25. queeems is designed with optimal user flexibility in mind. Therefore, there is need to initiate the packages’ operations with certain parameter values. The parameter settings that produced the results for this section are listed in the table below.

Parameter settings used to initiate the queeems analyses of saturation influenced by tree lengths. Readers should consult the function manual available with the package, for extensive discussions about the implications of the listed inputs.
Inference approach “A”
Discretised weight size 6
Proportion of weights to be \(<\) 1 0.40
Weight upper-bound 7.00
Bayes factor threshold 3.00
Tolerance for saturated sites 0.01 
# ><>< # Load package into current R session
library("queeems")
## Loading required package: Biostrings
## Loading required package: BiocGenerics
## Loading required package: generics
## 
## Attaching package: 'generics'
## The following objects are masked from 'package:base':
## 
##     as.difftime, as.factor, as.ordered, intersect, is.element, setdiff,
##     setequal, union
## 
## Attaching package: 'BiocGenerics'
## The following objects are masked from 'package:stats':
## 
##     IQR, mad, sd, var, xtabs
## The following objects are masked from 'package:base':
## 
##     anyDuplicated, aperm, append, as.data.frame, basename, cbind,
##     colnames, dirname, do.call, duplicated, eval, evalq, Filter, Find,
##     get, grep, grepl, is.unsorted, lapply, Map, mapply, match, mget,
##     order, paste, pmax, pmax.int, pmin, pmin.int, Position, rank,
##     rbind, Reduce, rownames, sapply, saveRDS, table, tapply, unique,
##     unsplit, which.max, which.min
## Loading required package: S4Vectors
## Loading required package: stats4
## 
## Attaching package: 'S4Vectors'
## The following object is masked from 'package:utils':
## 
##     findMatches
## The following objects are masked from 'package:base':
## 
##     expand.grid, I, unname
## Loading required package: IRanges
## Loading required package: XVector
## Loading required package: Seqinfo
## 
## Attaching package: 'Biostrings'
## The following object is masked from 'package:base':
## 
##     strsplit
# ><>< # Create useful summary function
infoests <- function(xdata){
    serr <- sd(xdata) / sqrt( length(xdata))
    avg <- mean( xdata )
    low <- avg  -  serr
    upp <- avg  +  serr
    outs <- c(low=low, avg=avg, upp=upp)
    return( outs )
}

# ><>< # Retrieve path to simulated tutorial data folder
datadir <- system.file("extdata/example/stemL/", package="queeems")

# ><>< # Indicate number of simulated data replicates
nreps <- 5

# ><>< # Set preferred Bayes factor threshold
bflimit <- 3

# ><>< # Set preferred saturated sites tolerance
ssprop <- 0.01

# Branch length
blent <- seq(0.03, 0.15, 0.03)
btags <- sprintf("%02.0f", blent*100)

# ><>< # Create data frames to save interesting outputs
cnames <- paste("bl", btags, sep="")
rnames <- c("low","avg","upp")
vnames <- paste("rep.", seq(1,nreps), sep="")
homoBF <- matrix(NA, nreps, length(btags), dimnames=list(vnames,cnames))
sbayes <- matrix(NA, nreps, length(btags), dimnames=list(vnames,cnames))
smrybx <- matrix(NA, 3, length(btags), dimnames=list(rnames,cnames))
cdnTmp <- aasTmp <- vector("numeric", 100*nreps)
nucTmp <- vector("numeric", 300*nreps)
nucent <- codent <- aasent <- smrybx

for(a1 in seq(1,length(btags))){
    # ><>< # Read merged molecular sequences data as text
    seqsname <- file.path(datadir, paste("seqs",btags[a1],".txt", sep=""))
    seqstext <- readLines(seqsname)

    # ><>< # Extract information about number of simulated replicates
    sbreaks <- which(seqstext == "  64  300")

    # ><>< # Analyse data replicates captured from merged data file
    for(a0 in seq(1,nreps)){
        nucentID <- seq(1+(a0-1)*300, a0*300)
        entropyID <- seq(1+(a0-1)*100, a0*100)

        # ><>< # Extract and temporarily save a sequence replicate
        init <- sbreaks[a0] + 1
        halt <- sbreaks[a0] + 128
        seqrep <- seqstext[seq(init,halt)]
        tempfile <- file.path(tempdir(), "iseqs.fasta")
        write.table(seqrep, file=tempfile, append=FALSE,
            quote=FALSE, row.names=FALSE, col.names=FALSE)

        # ><>< # Execute saturation analysis
        satest <- seqSaturation(tempfile, "A", 6, 0.4, 7, bflimit, ssprop)

        # ><>< # Extract Bayes factors
        homoBF[a0,a1] <- as.numeric( summary(satest)["fullBF",])
        sbayes[a0,a1] <- sum(BFs(satest) > bflimit)

        # ><>< # Execute codon entropy analysis
        tempentropy <- molentropy(tempfile, "codon", "shannon")
        cdnTmp[entropyID] <- sitentropies(tempentropy)

        # ><>< # Process nucleotide entropy analysis
        tempentropy <- molentropy(tempfile, "dna", "shannon")
        nucTmp[nucentID] <- sitentropies(tempentropy)

        # ><>< # Transform nucleotide to amino acid sequences
        dnaformat <- Biostrings::readDNAStringSet(tempfile)
        aaformat <- Biostrings::translate(dnaformat)
        Biostrings::writeXStringSet(aaformat, tempfile)

        # ><>< # Implement amino acid entropy analysis
        tempentropy <- molentropy(tempfile, "aa", "shannon")
        aasTmp[entropyID] <- sitentropies(tempentropy)
    }
    # ><>< # Summarise and save entropy values
    nucent[rnames,a1] <- infoests( nucTmp )[rnames]
    codent[rnames,a1] <- infoests( cdnTmp )[rnames]
    aasent[rnames,a1] <- infoests( aasTmp )[rnames]

    # ><>< # Provide analysis progress update
    message("Completed analyses of stem length 0.", btags[a1])
}
## Completed analyses of stem length 0.03
## Completed analyses of stem length 0.06
## Completed analyses of stem length 0.09
## Completed analyses of stem length 0.12
## Completed analyses of stem length 0.15
# ><>< # Site-wise Bayes factors peak
print( head( BFs(satest)))
## [1] 1.957368 2.039552 1.311460 1.100639 1.297898 1.487825
# ><>< # Saturated site counts
print(sbayes)
##       bl03 bl06 bl09 bl12 bl15
## rep.1    0    1    2    2    3
## rep.2    1    0    1    1    1
## rep.3    0    1    0    3    4
## rep.4    0    1    2    1    2
## rep.5    1    2    1    0    2
# ><>< # Overall Bayes factor estimates
print(homoBF)
##         bl03   bl06   bl09   bl12    bl15
## rep.1 0.3660 0.9109 1.4864 1.4864  3.7668
## rep.2 0.9109 0.3660 0.9109 0.9109  0.9109
## rep.3 0.3660 0.9109 0.3660 3.7668 13.5471
## rep.4 0.3660 0.9109 1.4864 0.9109  1.4864
## rep.5 0.9109 1.4864 0.9109 0.3660  1.4864

Next, I present visual summaries of queeems analyses output. All the Bayes factors are based on the alternative hypothesis that the corresponding sequences are saturated. Therefore, it is expected that the Bayes factor estimates will increase as the tree length increases. Tree length is calculated as the distance from the origin of the rooted balanced bifurcating phylogeny to the tip of the terminal branches. The increasing pattern in Figure A, consequently demonstrates that the outputs from queeems are reliable.

# ><>< # Function that makes plotting easier
tfunc <- function(i, pty, clr, cx, mat, agl=90){
    arrows(i, mat["low",i], i, mat["upp",i], .1, agl, 3, colors()[clr], lwd=3)
    points(i,mat["avg",i],pch=pty,lwd=2,col=colors()[clr],cex=cx,bg="white")
    return(0)
}

# ><>< # Summarise overall Bayes factor values
bfsumm <- apply(homoBF,2,infoests)

par(mar=c(4.2,4.2,0.2,0.2))
plot(0, 0, col="white", bty="n", xlab="", ylab="",
    xlim=c(1.0,5.0), ylim=c(0.0,7.0), xaxt="n", yaxt="n")
phold <- vapply(seq(1,5), tfunc, 15, 50, 2.5, bfsumm, FUN.VALUE=0)
axis(1, seq(1,5), seq(3,15,3)/100*7, tck=-.01, lwd=2, padj=-.2, cex.axis=1.5)
axis(2, las=1, tck=-0.01, lwd=2, cex.axis=1.5, hadj=0.2)
text(1.3, 6.5, "A", cex=2.5, font=2)
mtext("Bayes factor", 2, 2.2, cex=2)
mtext("Tree Length", 1, 2.4, cex=2)
box(lwd=2)

The inference from Figure A is interrogated further by plotting the Shannon entropy estimates, presented in Figure A.1 and the site-wise Bayes factors, presented in Figure A.2. To ensure accurate comparisons, the entropy estimates are scaled with the maximum possible values. The maximum Shannon entropy for a sequence with n possible bases is \(-\log(1/n)\). For nucleotide, sense codon and amino acid bases, n = 4, n = 61 and n = 20, respectively.

The points in the Figure A.1 are the average and the arrow widths are proportional to the standard errors. The plot justifies the choice of codon as a robust unit for quantifying saturation. Amino acid appear to be moving faster to saturation (that is, closer to one) for shorter lengths. As the lengths get longer, nucleotide bases appears to close the gap with amino acids and most probably surpass it. The entropy analyses presented here were obtained by applying the popular Shannon14 method on the base frequencies. Other counting metrics are accessible in queeems. These include: the cncsentropy and the codondifferindex functions. The package is also designed in a way that permits users to opt for Renyi entropy calculation method, if preferred. Readers may consult the help pages of the respective functions for further details.

The histogram in Figure A.2 depicts the distribution of the site-wise Bayes factors. For brevity, only the output for the fifth replicate for the 1.05 tree length analysis is shown. As expected for non-saturated sequences, the distribution is positively skewed with majority of the sites returning Bayes factors less than 3.0. This implies that the corresponding genetic sequence is most likely not saturated.

# ><>< # Scale entropy estimates by the maximum value
eScaledNuc <- nucent / (-log(1/4))
eScaledCodon <- codent / (-log(1/61))
eScaledAmino <- aasent / (-log(1/20))

# ><>< # Plot distributions of entropy estimates
par(mar=c(4.2,4.2,0.2,0.2))
bases <- c("Nucleotide","Codon","Amino acid")
plot(0, 0, col="white", bty="n", xlab="", ylab="",
    xlim=c(1.0,5.0), ylim=c(.19,0.57), xaxt="n", yaxt="n")
phold <- vapply(seq(1,5), tfunc, 0, 17, 1.5, eScaledNuc, FUN.VALUE=0)
phold <- vapply(seq(1,5), tfunc, 1, 60, 1.5, eScaledCodon, FUN.VALUE=0)
phold <- vapply(seq(1,5), tfunc, 8, 74, 1.5, eScaledAmino, FUN.VALUE=0)
axis(1, seq(1,5), seq(3,15,3)/100*7, tck=-.01, lwd=2, padj=-.5, cex.axis=1.5)
text(rep(3.7,3), c(.27,.24,.21)-.002, bases, cex=1.5, pos=4)
axis(2, las=1, tck=-0.01, lwd=2, cex.axis=1.5, hadj=0.7)
points(rep(3.6,3), c(.27,.24,.21), cex=2.5, lwd=2,
    pch=c(0,1,8), col=colors()[c(17,60,74)])
text(3.8, 0.30, "Base", cex=2, font=2)
mtext("Scaled Entropy", 2, 2.4, cex=2)
text(1.3, 0.54, "A.1", cex=2.5, font=2)
mtext("Tree Length", 1, 2.2, cex=2)
box(lwd=2)

# ><>< # Site-wise Bayes factors from replicate 5 of length 1.05
par(mar=c(4.2,4.2,0.2,0.2))
plot(0, 0, col="white", bty="n", xlab="", ylab="",
    xlim=c(.6,4.4), ylim=c(.03,0.85), xaxt="n", yaxt="n")
mtext(c("Site-Wise Bayes Factors","Density"), c(1,2), 2.4, cex=2)
axis(2, las=1, tck=-0.01, lwd=2, cex.axis=1.5, hadj=0.7)
hist(BFs(satest), freq=FALSE, add=TRUE, xaxt="n",
    yaxt="n", main="", col=colors()[17], lwd=2)
axis(1, tck=-.01, lwd=2, padj=-.5, cex.axis=1.5)
text(4.1, 0.81, "A.2", cex=2, font=2)
box(lwd=2)

Natural selection analyses

The parameters input for the analyses in this section are the same as those used for the tree length analyses. The tree length was however fixed as 0.7 for all simulations. Non-synonymous selection coefficient variance (\(\sigma_{`n`}^{2}\)) is the variable in this section and values considered are, (0.1, 0.2, 0.3, 0.4, 0.5). Figure B contains summaries of the corresponding overall sequence Bayes factors. The points represent the averages and the arrows are proportional to the standard errors estimated over the five data replicates analysed for each \(\sigma_{`n`}^{2}\) value. The analytical estimates of the ratio of non-synonymous to synonymous substitution rates (\(\mathrm{d}N/\mathrm{d}S\)) were obtained along with the sequences from scoup, where they were calculated using expressions from Spielman and Wilke15.

# ><>< # Retrieve path to simulated tutorial data folder
datadir <- system.file("extdata/example/varNS/", package="queeems")

# Non-synonymous variance value
nsvary <- seq(1, 5) / 10
ntags <- sprintf("%.0f", nsvary*10)

# ><>< # Create data frames to save interesting outputs
cnames <- paste("ns", ntags, sep="")
rnames <- c("low","avg","upp")
homoBF <- matrix(NA, nreps, length(ntags), dimnames=list(NULL,cnames))
sbayes <- matrix(NA, nreps, length(ntags), dimnames=list(NULL,cnames))

for(a1 in seq(1,length(ntags))){
    # ><>< # Read merged molecular sequences data as text
    seqsname <- file.path(datadir, paste("gdata",ntags[a1],".txt",sep=""))
    seqstext <- readLines(seqsname)

    # ><>< # Extract information about number of simulated replicates
    sbreaks <- which(seqstext == "  64  300")

    # ><>< # Analyse data replicates captured from merged data file
    for(a0 in seq(1,nreps)){

        # ><>< # Extract and temporarily save a sequence replicate
        init <- sbreaks[a0] + 1
        halt <- sbreaks[a0] + 128
        seqrep <- seqstext[seq(init,halt)]
        tempfile <- file.path(tempdir(), "iseqs.fasta")
        write.table(seqrep, file=tempfile, append=FALSE,
            quote=FALSE, row.names=FALSE, col.names=FALSE)

        # ><>< # Execute saturation analysis
        satest <- seqSaturation(tempfile, "A", 6, 0.4, 7, bflimit, ssprop)

        # ><>< # Extract Bayes factors
        homoBF[a0,a1] <- as.numeric( summary(satest)["fullBF",])
        sbayes[a0,a1] <- sum(BFs(satest) > bflimit)
    }
    # ><>< # Provide analysis progress update
    message("Completed analyses for \U03c3\U00b2 = 0.", ntags[a1])
}
## Completed analyses for σ² = 0.1
## Completed analyses for σ² = 0.2
## Completed analyses for σ² = 0.3
## Completed analyses for σ² = 0.4
## Completed analyses for σ² = 0.5
# ><>< # Overall log(Bayes factor) estimates
print( log(homoBF))
##              ns1         ns2         ns3        ns4         ns5
## [1,] -1.00512195 -0.09332216 -0.09332216  2.6061725 -0.09332216
## [2,]  1.32622583  0.39635709 -0.09332216  0.3963571 -0.09332216
## [3,] -0.09332216  4.14920673  2.60617250  1.3262258  4.14920673
## [4,] -0.09332216 -1.00512195 -1.00512195  0.3963571  2.60617250
## [5,] -0.09332216 -0.09332216  0.39635709 -1.0051219  0.39635709
# ><>< # Summarise overall Bayes factor and dN/dS values
bfsumm <- apply(log(homoBF),2,infoests)
dndsPath <- file.path(datadir, "dnds.csv")
dndses <- read.table(dndsPath, sep=";", header=TRUE)

# ><>< # Facilitate super-imposition of dN/dS and BFs
transplot <- function(x, xrng, newrng){
    xmin <- xrng[1]
    ymin <- newrng[1]
    xdiff <- diff(xrng)
    ydiff <- diff(newrng)
    newx <- (((x - xmin) / xdiff) * ydiff) + ymin
    return(newx)
}
newaxs <- seq(4,9)
oldaxs <- transplot(newaxs, c(5,9), c(-.4,2.3))
newDnDs <- transplot(dndses, c(5,9), c(-.4,2.3))
dndsumm <- apply(newDnDs, 2, infoests)

# ><>< # Add summaries of the BF values to plot
par(mar=c(4.4,4.2,0.2,3.0))
plot(0, 0, col="white", bty="n", xlab="", ylab="",
    xlim=c(1.0,5.0), ylim=c(-.4,2.3), xaxt="n", yaxt="n")
phold <- vapply(seq(1,5), tfunc, 15, 50, 2.5, bfsumm, FUN.VALUE=0)
axis(1, seq(1,5), nsvary, tck=-.01, lwd=2, padj=-.5, cex.axis=1.5)
axis(2, las=1, tck=-0.01, lwd=2, cex.axis=1.5, hadj=0.7)
mtext(expression(sigma[n]^2), 1, 3.4, cex=2)
mtext("Log(Bayes factor)", 2, 2.5, cex=2)
text(1.2, 2.2, "B", cex=2.5, font=2)

# ><>< # Include summaries of the dN/dS values
phold <- vapply(seq(1,5), tfunc, 23, 17, 2, dndsumm, 60, FUN.VALUE=0)
axis(4, oldaxs, newaxs, las=1, tck=-0.01, lwd=2, cex.axis=1.5, hadj=0.7)
mtext("dN/dS", 4, 2.0, cex=2)

# ><>< # Add legend to plot
text(c(2,2), c(2.2,2.0), c("BF","dN/dS"), cex=1.8, pos=4)
arrows(1.7, 2.21, 2.0, 2.21, .1, 90, 3, colors()[50], 1, 3)
arrows(1.7, 2.0, 2.0, 2.0, .1, 60, 3, colors()[17], 1, 3)
points(c(1.85,1.85), c(2.21,2.0), cex=2, bg="white",
    lwd=3, pch=c(15,23), col=colors()[c(50,17)])
box(lwd=2)

It is illustrated in Figure B that higher values of \(\sigma_{`n`}^{2}\) translates to higher magnitude of positive natural selection pressure, as measured by \(\mathrm{d}N/\mathrm{d}S\). Therefore, as expected, the magnitude of the corresponding Bayes factors mostly increased accordingly. This inference further supports the authenticity of the inferences generated by the queeems package.

Conclusion

I demonstrated that queeems avails increased flexibility that enables researchers to investigate genetic sequence saturation with better insights. The choice of the queeems parameters applied for the applications presented here are arbitrary. I appreciate that separate studies may require different values depending on the tolerance afforded by the conditions of the investigations being undertaken. A study about the sensitivity of the parameters is nearing completion.

Citation

A more appropriate citation will be provided for the package in due course. In the meantime, kindly cite the package as, Sadiq, H. 2026. “queeems: Quantify the Extent of Evolutionary Evidence in Molecular Sequences”. Bioconductor R Package.

Version Information

The queeems package adopts versioning requirement of the Bioconductor16 platform. Its first version (v1.0.0) includes one Bayesian saturation analysis method. Subsequent versions (>= v2.x.x) of the package should contain more methods as described in the seqSaturation manual page.

Session info

The output of sessionInfo() from the computer where this file is generated is provided below.

## R version 4.6.0 (2026-04-24)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## time zone: Etc/UTC
## tzcode source: system (glibc)
## 
## attached base packages:
## [1] stats4    stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
## [1] queeems_1.0.0       Biostrings_2.80.0   Seqinfo_1.2.0      
## [4] XVector_0.52.0      IRanges_2.46.0      S4Vectors_0.50.0   
## [7] BiocGenerics_0.58.0 generics_0.1.4      BiocStyle_2.40.0   
## 
## loaded via a namespace (and not attached):
##  [1] crayon_1.5.3        cli_3.6.6           knitr_1.51         
##  [4] rlang_1.2.0         xfun_0.57           gtools_3.9.5       
##  [7] jsonlite_2.0.0      buildtools_1.0.0    htmltools_0.5.9    
## [10] maketools_1.3.2     sys_3.4.3           sass_0.4.10        
## [13] rmarkdown_2.31      grid_4.6.0          evaluate_1.0.5     
## [16] jquerylib_0.1.4     fastmap_1.2.0       yaml_2.3.12        
## [19] lifecycle_1.0.5     BiocManager_1.30.27 compiler_4.6.0     
## [22] lattice_0.22-9      digest_0.6.39       R6_2.6.1           
## [25] Matrix_1.7-5        bslib_0.10.0        tools_4.6.0        
## [28] cachem_1.1.0

References

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  1. R Core Team (2025)↩︎

  2. Gentleman et al. (2004)↩︎

  3. Xia et al. (2003)↩︎

  4. Manuel et al. (2025)↩︎

  5. Struck (2014)↩︎

  6. Swofford (2002)↩︎

  7. Lyons-Weiler et al. (1996)↩︎

  8. Philippe (1993)↩︎

  9. Fleming and Alberto Valero-Gracia (2023)↩︎

  10. Sadiq and Martin (2026)↩︎

  11. Xia et al. (2003)↩︎

  12. Manuel et al. (2025)↩︎

  13. Shannon (1948)↩︎

  14. Shannon (1948)↩︎

  15. Spielman and Wilke (2015)↩︎

  16. Gentleman et al. (2004)↩︎