Wurm lab

Leukerbad Spring School Bioinformatics and Population Genomics


Practical: RNA-seq analysis for population genomics

Julien Roux, version 1, May 2016

Schedule

Introduction

Today you will pursue the analysis of Bou Sleiman et al. Drosophila melanogaster data. Unless you managed to map all samples yesterday (:clap:), use the pre-processed Kallisto results located in the ~/data/rnaseq/kallisto/SRR* folders. The analysis will be going through the following steps:

In the interest of time, most of the R commands are given fully, and can simply copy-paste them. I encourage you to read the commands fully and try to understand what was done. Try and modify some parameters, or ask the assistants if something is not clear.

Read and format the metadata

Don't forget to create today's working directory:

mkdir -p ~/2016-06-02-rnaseq/results/

Launch R in the console or use Rstudio. Then:

## path to the data
dataFolder <- "~/data/rnaseq/"
## Set up the working directory
setwd("~/2016-06-02-rnaseq/results/")

You will first read and format the experiment metadata using the GEOquery package (http://www.bioconductor.org/packages/release/bioc/vignettes/GEOquery/inst/doc/GEOquery.html). The metadata are located in the GSE59411_series_matrix.txt file that was pre-downloaded on the GEO page of the experiment. Metadata can also be read from the GSE59411_family.soft file, or by passing the experiment name to the GEOquery package (beware, the format of the resulting R object will slightly differ depending on the approach used). The following steps are a bit boring, but try to understand what was done:

## Should not be needed:
## source("http://bioconductor.org/biocLite.R")
## biocLite("GEOquery")
library(GEOquery)
## Read the metadata file
gse <- getGEO(filename=file.path(dataFolder, "GSE59411_series_matrix.txt"))
## Extract the metadata, and only retain some of the columns
samples <- pData(phenoData(gse))[,c(1,2, 8:12, 45)]
## Extract the SRA ID of samples by parsing the FTP URL
samples$library_accession <- unlist(
  lapply(strsplit(as.character(samples$supplementary_file_2), split="/"), tail, n=1))
## Read the ENA metadata, which includes the run ID (SRR...)
samplesENA <- read.table(file.path(dataFolder, "SRP044339.txt"), h=T, sep="\t")
## Merge both data frames
samples <- merge(samples, samplesENA, by.x="library_accession", by.y="experiment_accession")
## Filter and rearrange the columns
samples <- samples[,c(2, 4:8, 3, 1, 14)]
## Rename the columns
names(samples)[4] <- "treatment"
names(samples)[5] <- "dgrp_line"
names(samples)[6] <- "resistance"
## Simplify the levels of experimental factors
samples$treatment <- gsub("treatment: ", "", samples$treatment)
samples$dgrp_line <- gsub("dgrp line: ", "", samples$dgrp_line)
samples$resistance <- gsub("resistance: ", "", samples$resistance)
## Visualize the formatted metadata table
head(samples)
dim(samples)

Question
According to the metadata table, how many experimental factors of interest do you identify in this experiment? Draw a simple diagram to represent the experimental design.

For simplicity, we will ignore the strain effect (dgrp_line), and consider the four susceptible and the four resistant lines as simple biological replicates.

Import the transcripts expression levels and sum them at the gene level

You will use the Mike Love's tximport package to directly parse Kallisto results files and sum expression at the gene-level (https://github.com/mikelove/tximport). As tximport was not yet available on R/Bioconductor 3.2 installed on the virtual machine (it is now available in release 3.3), you will install it from the source code, that is already in the data folder:

install.packages(file.path(dataFolder, "./tximport-master"), repos = NULL, type="source")
library(tximport)

## Path to Kallisto result files
files <- file.path(dataFolder, "kallisto", samples$run_accession, "abundance.tsv")
names(files) <- samples$title
## Check that all files are present in the data folder
all(file.exists(files))

To sum up transcript expression levels, tximport needs a data.frame with a transcript ID column and a gene ID column. You will retrieve this information from the Ensembl database via the Biomart webservice. Biomart queries require the specification of:

1) a dataset to use

2) a list of filters to select specific Ensembl genes (not needed in our case)

3) a list of attributes to output.

It is possible to build the Biomart query from the web interface (the needed query would be obtain by specifying the following dataset and attributes), but there is a Bioconductor package called biomaRt that allows to obtain results directly into R:

library(biomaRt)
## Choose D. melanogaster dataset in Ensembl release 84
mart <- useMart("ENSEMBL_MART_ENSEMBL", dataset = "dmelanogaster_gene_ensembl", host="mar2016.archive.ensembl.org")
## retrieve the mapping between transcript and gene IDs
transcript2gene <- getBM(attributes=c("ensembl_transcript_id", "ensembl_gene_id"), mart=mart) 
## Sort by gene ID
transcript2gene <- transcript2gene[order(transcript2gene$ensembl_gene_id), ]
head(transcript2gene)

Tip
Tip: specifying the host argument in the biomaRt query allows to choose which Ensembl release you wish to work with (in this case, mar2016.archive.ensembl.org redirects to release 84). This is very helpful to make your code reproducible.

You will now import Kallisto results:

txi <- tximport(files, type = "kallisto", tx2gene = transcript2gene, countsFromAbundance="scaledTPM")
## Visualize the created txi object:
lapply(txi, head)

Question
What are abundances? What are "counts" in this particular case? Why are length estimates not integers (remember the note about Kallisto --bias option in yesterday practical)?

I suggest to have a look at this interesting paper:

Soneson C, Love M and Robinson M. Differential analyses for RNA-seq: transcript-level estimates improve gene-level inferences. F1000Research. 2015;4(1521) (http://f1000research.com/articles/4-1521/v2). A PDF of the paper is located in the ~/data/papers/ folder.

In this paper, Soneson and colleagues show that gene-level abundance estimates offer advantages over transcript-level analyses. They are more accurate, more stable, more interpretable, and allow better control on false positives in the presence of differential isoform usage. For this purpose, they introduced the "scaledTPM" values, which are obtained by summing the transcript-level TPMs by gene, and multiplying them with the total library size in millions. ScaledTPM values are artificial values, transforming underlying abundance measures to the scale of read counts. This allows to incorporate the information provided by the sequencing depth, and work with RNA-seq differential expression tools that were developed to use read counts, while controlling for the length of transcripts expressed for each gene in each given condition.

Data normalization

You will import the gene expression data into a edgeR DGE object and obtain normalization factors for each sample. The default normalization method in edgeR is called TMM. It was previously shown to perform quite well, while not too stringent.

library(edgeR)
library(limma)
y <- DGEList(txi$counts)
y <- calcNormFactors(y)
y$samples

Question
What is the meaning of each column in this data frame?

You will now look at the data distribution in the DGE object. The cpm function returns counts per million reads. Because you used scaledTPMs instead of real counts, this is equivalent to the initial TPMs used to create the scaledTPMs. But these are additionally normalized by dividing each samples values by the TMM normalization factor:

## Load a nice color palette of 9 + 8 colors to be used for plots
library(RColorBrewer)
myPalette <- c(brewer.pal(9, "Set1"), brewer.pal(8, "Set2"))
## Plot CPM distribution
plotDensities(cpm(y), col=myPalette[1:16], legend="topright")
## Try with logged CPM
unfilteredExpr <- cpm(y, log=T)
?cpm ## note that a small value is added to the counts if log=T
plotDensities(unfilteredExpr, col=myPalette[1:16], legend="topright")

Question
What do you observe? How are the data distributed? Why are there two modes? What would be a good cutoff to discriminate the two modes?

It is a good practice to filter out the genes that are not expressed, or very lowly expressed. This alleviates the multiple testing burden, and anyway there is very little power to detect differential expression for these genes. It is of course better to filter these genes after data normalization, before differential expression testing. The choice of a criterion to filter genes is arbitrary, but it has to make sense given the distribution of the data. You can for example put a cutoff on the summed expression of each gene across samples. Another apporach I tend to prefer, is to keep only the genes that pass a cutoff in at least n samples, where n is at least the number of biological replicates in each experimental condition. This ensures that almost all genes are seen expressed in at least one condition.

Warning
Unless you know what you are doing, it is considered a bad practice to filter genes based on their variance prior to differential analysis! This can seriously bias your results.

To do
Indicate below the cutoff expression that was chosen in the previous question

cutoff <- ... 
## How many genes pass this cutoff in each sample:
summary(unfilteredExpr > cutoff)
## For each row return the number of expression values above the cutoff
## This give sthe number of samples in which each gene is expressed above the cutoff 
numSamplesWithExpression <- apply(unfilteredExpr, 1, function(x){ return(sum(x > cutoff)) })
## Plot this as a histogram
hist(numSamplesWithExpression)
## Most genes are either expressed in all samples, or in no sample!
## You can for example retain all genes expressed in at least half of the samples:
selectedGenes <- names(which(numSamplesWithExpression >= 8))
length(selectedGenes)

You will now rebuild a new DGE object using only selected genes, and renormalize it:

y <- DGEList(txi$counts[selectedGenes, ]) 
y <- calcNormFactors(y)
filteredExpr <- cpm(y, log=T)
plotDensities(filteredExpr, col=myPalette[1:16], legend="topright")

Question
What has changed now? Do you like the distribution of expression levels and their normalization?

Let's look at the distribution of samples depending on the 2 experimental factors:

plotDensities(filteredExpr, group=samples$resistance, col=myPalette[1:2])
plotDensities(filteredExpr, group=samples$treatment, col=myPalette[3:4])

Question
Are there systematic differences in the distribution of expression levels that are linked to one or the other experimental factor? Can you guess which experimental factor is likely to have the strongest effect on expression levels in this experiment?

Data clustering

PCA

You will perform a principal component analysis (PCA) on the normalized data using the prcomp R command. Input data of the PCA need to be ~normally distributed, so you will use the logged-CPM values previously plotted.

pca <- prcomp(t(filteredExpr), scale = T)
plot(pca)
summary(pca)
loadings <- pca$rotation
scores <- pca$x

Question
What does the scale=T argument means? How much variance is explained by the first 2 principal components? The first 4 principal components? What the scores and loadings?

Plot the samples projected onto the first two principal components.

plot(scores[,1], scores[,2])
## OK, here is a bit nicer plot ;)
plot(scores[,1], scores[,2], 
     xlab=paste0("PC1: ", round(summary(pca)$importance[2,1],3)*100, "% variance explained"), 
        ## indicate the % variance explained by PC1
     ylab=paste0("PC2: ", round(summary(pca)$importance[2,2],3)*100, "% variance explained"), 
        ## indicate the % variance explained by PC2
     pch=as.numeric(as.factor(samples$treatment))+15,
        ## points shape according to treatment
     col=myPalette[as.numeric(as.factor(samples$resistance))], 
        ## points color according to susceptibility
     xlim=c(min(scores[,1]), max(scores[,1])) ,
     ylim=c(min(scores[,2]), max(scores[,2])+(max(scores[,2])-min(scores[,2]))/4) 
        ## Let a bit of room on top of the plot for legend
)
## Plot legends
legend("topleft", legend=levels(as.factor(samples$treatment)), pch=16:17)
legend("topright", legend=levels(as.factor(samples$resistance)), pch=c(16,16), col=myPalette[1:2])

Question
What experimental factor does correlate with PC1? What is special about PC2? It could be useful to add sample names with the following command (the + 5 allows to put the labels next to the points and not on the points)

text(scores[,1], scores[,2] + 5, samples$title)

Tip
When labels get too messy, it can be nice to only label the interesting points. The command identify(scores[,1], scores[,2], samples$title) allows you to click on the points to reveal their labels. You can also influence the way the labels are placed by clicking slightly above/below/left/right of a point. Press escape to exit the clicking mode.

To do
Plot the samples projected onto PC2 and PC3, then PC3 and PC4. Do you observe segregation of the points by any experimental factor? Are these observations consistent with your previous prediction of the experimental factor with the strongest effect on expression levels?

Heatmap (skip this part if timing is tight)

Another way to visualize the data is to plot a heatmap of expression levels, along with a dendrogram obtained by hierarchical clustering of the samples. The heatmap R function allows you to perform this, but the heatmap.2 from the gplots package offers more possibilities. There a lot more packages and heatmap functions to explore if you have particular needs. You will first plot a heatmap using a random selection of 100 genes:

library(gplots)
## create a gradient of 100 colors going from light blue to dark blue
colors <- colorRampPalette(c(brewer.pal(9, "Blues")[1],brewer.pal(9, "Blues")[9]))(100)
## select randomly 100 genes and extract their expression
selectedExpression <- filteredExpr[sample(1:length(filteredExpr[,1]), 100),]
## Plot the heatmap
heatmap.2(selectedExpression, scale="none", col = colors, margins = c(14, 6), trace='none', denscol="white")

Question
What are the rows and the left tree representing? What are the columns and the top tree representing? What does the color intensity mean? Try to replot the heatmap with a new random selection of 100 genes. Is the clustering stable?

It is difficult to include the expression of all genes to create a readable heatmap. An alternative is to calculate the matrix of pairwise correlation coefficients across all samples and plot a heatmap of this matrix. In addition, the ColSideColors and RowSideColors arguments allow to better visualize the experimental factors of each sample:

allCors <- cor(filteredExpr, method="spearman", use="pairwise.complete.obs")
heatmap.2( allCors, scale="none", col = colors, margins = c(16, 12), trace='none', denscol="white", 
           RowSideColors=myPalette[1:2][as.integer(as.factor(samples$resistance))], 
           ColSideColors=myPalette[3:4][as.integer(as.factor(samples$treatment))])

Question
What are the rows and the columns representing now? What does the color intensity mean? Is the clustering pattern consistent with the PCA? Do you see a manifestation of the DGRP line effect on the heatmap?

Differential expression analysis

Some background

There are several softwares to test for differential expression. DESeq2 and edgeR are very good, but here you will use the limma-voom method, introduced in this paper:

Law C, Chen Y, Shi W, Smyth G. voom: precision weights unlock linear model analysis tools for RNA-seq read counts. Genome Biology. 2014;15(2):R29 (http://genomebiology.biomedcentral.com/articles/10.1186/gb-2014-15-2-r29). A PDF of the paper is located in the ~/data/papers/ folder.

While DESeq2 and edgeR work directly on count distributions, using a negative binomial modeling framework, limma-voom works on the log-transformed CPMs, which are normally distributed data. This is very useful because the theory behind of normal distributions is more tractable. Notably, a large range of statistical methods that were developped for microarray analysis can be used on such normally distributed data. For example, the widely used limma package uses an empirical Bayes approach, which borrows information across genes to adjust variance estimates when sample sizes are small. Limma is a very comprehensive package, able to deal with experiments with complex designs.

A problem of the log-transformation is that it does not yield stable variances (i.e., there is heteroscedasticity). The variability of genes with lower expression is lower than those with high expression, This is easily observable when plotting one sample versus a biological replicate:

## The color argument was set up to plot transparent black points 
plot(filteredExpr[,5], filteredExpr[,6], pch=16, col=rgb(0,0,0,0.2))

To deal with this problem, limma-voom models the mean-variance relationship observed in the data. This trend is incorporated into a precision weight for each individual normalized observation, which are proportional to the expression levels (see https://en.wikipedia.org/wiki/Least_squares#Weighted_least_squares). These weights can be used in limma during its linear modeling step. Limma-voom was shown to perform equally well compared to tools based on the negative-binomial distribution modeling, or even better when the sequencing depths are different across samples.

An additional interesting possibility offered by the limma package is the possibility to adjust the limma-voom precision weights to deal with variations in sample quality, frequently encountered in RNA-seq experiments. Removal of high variation samples reduces noise but leads to a loss of power. A compromise is to use all samples, but to down-weight the observations from more variable samples. This is implemented in the voomWithQualityWeights function, described in the following paper:

Liu R, et al. Why weight? Modelling sample and observational level variability improves power in RNA-seq analyses. Nucleic Acids Res. 2015;43(15):e97 (http://nar.oxfordjournals.org/content/43/15/e97.long). A PDF of the paper is located in the ~/data/papers/ folder.

Question
Following the PCA results, do you think this procedure will be beneficial?

Implementation

## Limma-voom requires the specification of a design matrix. 
## It is simpler to create a single factor made of the combination of the 2 factors of interest: 
## susceptibility and treatment
condition <- factor(paste(samples$treatment, samples$resistance, sep="."))
## Build the design matrix:
design <- model.matrix(~ 0 + condition) 
## The "~ 0 + ..." syntax is optional, but it will later allow an easier specification of the 
## contrasts for differential expression
## Simplify design matrix column names:
colnames(design) <- gsub("condition", "", colnames(design))
## Apply the limma-voom method:
v <- voomWithQualityWeights(y, design, plot=T)

Question
What are the generated plots indicating? Which sample(s) are the most down-weighted?

Linear modeling and extraction of interesting contrasts

Please refer to the very nice user guide for limma for details on the analysis, or to adapt this pipeline to analyze your own data (http://www.bioconductor.org/packages/release/bioc/vignettes/limma/inst/doc/usersguide.pdf)

fit <- lmFit(v)

The next step is to specify the contrasts of interest in a contrast matrix. You can choose the names of the contrasts yourself (for example see below treatmentInS). To specify what the contrast will be comparing, you need to build a linear combinations of the design matrix column names. This means you need to choose among the following 4 terms: Challenged.Resistant, Unchallenged.Resistant, Challenged.Susceptible and Unchallenged.Susceptible. I have specified below the 3 following interesting contrasts:

To do
Following the same logic, please complete the R command to add the contrasts for:

You can now extract the lists of genes differentially expressed for each contrast, at a 10% FDR.

fit2 <- contrasts.fit(fit, cont.matrix)
fit2 <- eBayes(fit2)
results <- decideTests(fit2, p.value=0.1) 
summary(results)

Question
What do you observe? Is it consistent with the PCA? What is surprising?

You can see how many genes are differentially expressed in common between 2 contrast using Venn diagrams:

vennDiagram(results[,c(1,2)])
vennDiagram(results[,c(2,5)])
vennDiagram(results[,c(2,6)])

Question
What can you conclude about the overlap between lists of DE genes. Do these results make sense?

Extraction of differentially expressed genes

To extract the lists of differentially expressed genes, the coef argument is needed. It corresponds to the column number of the contrast matrix.

## Treatment:
treatmentGenes <- topTable(fit2, coef=3, p.value=0.1, number=Inf, sort.by="P")
## visualize the top 100 genes
selectedExpression <- filteredExpr[rownames(treatmentGenes)[1:100],]
heatmap.2(selectedExpression, scale="none", col = colors, margins = c(14, 6), trace='none', denscol="white", 
          ColSideColors=myPalette[3:4][as.integer(as.factor(samples$treatment))])
## Resistance:
resistanceGenes <- topTable(fit2, coef=6, p.value=0.1, number=Inf, sort.by="P")
## visualize all DE genes
selectedExpression <- filteredExpr[rownames(resistanceGenes),]
heatmap.2(selectedExpression, scale="none", col = colors, margins = c(14, 6), trace='none', denscol="white", 
          ColSideColors=myPalette[1:2][as.integer(as.factor(samples$resistance))])

Question
How does the clustering looks like when only resistance genes are taken into account? What does it tell you?

Bonus: characterization of differentially expressed genes

Manually

Because there are relatively few genes differentially expressed between resistant and susceptible lines, it is possible to look them up in reference databases (Ensembl, Uniprot, Flybase). But this long, subjective, and because there are a lot of uncharacterized genes, it is often frustrating...

GO enrichment test

To get an objective view of what are the genes in a long lists of genes, and to characterize their function, Gene Ontology (GO) enrichment analyses are useful. For each GO category (a group of genes sharing the same function, involved in the same process, or located in the same cellular compartment) it is possible to test whether the proportion of DE genes is higher then expected. You can do this with the topGO package.

Warning
A word of caution for the use of such tools:

library(topGO)

## topGO needs a vector with 0 or 1 values depending if a gene is DE or not
geneList <- rep(0, times=length(rownames(results)))
names(geneList) <- rownames(results)
DEGenes <- row.names(treatmentGenes)
geneList[DEGenes] <- 1
geneList = as.factor(geneList)
summary(geneList)

## You then need a mapping of genes to the GO categories. This can be retrieved from Ensembl using biomaRt, 
## or using the Drosophila melanogaster annotation package in Bioconductor
## If the annotation package is not installed: 
## source("http://bioconductor.org/biocLite.R")
## biocLite("org.Dm.eg.db")
library(org.Dm.eg.db)
## Build the topGO object for biological process ontology
BPdata <- new("topGOdata",
              ontology="BP",
              allGenes = geneList,
              nodeSize = 5,
              annot = annFUN.org,
              mapping = "org.Dm.eg.db",
              ID = "Ensembl")
resultBP <- runTest(BPdata, algorithm = "classic", statistic = "fisher")
myTable <- GenTable(BPdata, classic = resultBP, topNodes=length(BPdata@graph@nodes), numChar=100)
head(myTable)

To do
If you have time you can run the GO enrichment test on the molecular function (ontology="MF") or the cellular component ontologies (ontology="CC").

Question
What are the top categories enriched for genes DE with treatment? Is it consistent with what is reported in the original paper?

Tip
TopGO includes the possibility to use the p-values or scores for all genes, in order to perform the enrichment test without having to specify an arbitrary FDR cutoff to call genes differentially expressed or not. Several decorrelation algorithms are also implemented, which give less redundant, and more precise GO categories in the results. Refer to the topGO documentation for further info.

Question
Repeat the analysis with the weight algorithm (algorithm = "weight"), and observe the difference in results.

topAnat enrichment test

It is possible to perform a similar ontology enrichment, but on the fly anatomical ontology instead of Gene Ontology. Genes are mapped to a tissue if some expression was detected in this tissue. With the Bgee database team (http://bgee.org), I have developped a tool called BgeeDB allowing to do this, based on the topGO package algorithm. It is available in Bioconductor (release 3.3), or on GitHub https://github.com/BgeeDB/BgeeDB_R. I encourage you to try in addition to the classical GO enrichment tests, it gives very interesting results! A graphical interface is also available at http://bgee.org/?page=top_anat#/.

## As BgeeDB was not yet available on R/Bioc 3.2, install it from the source code, already in the data folder
install.packages("tidyr")
install.packages(file.path(dataFolder, "./BgeeDB-master"), repos = NULL, type="source")
library(BgeeDB)

## Loading calls of expression. This requires an internet connection
myTopAnatData <- loadTopAnatData(species=7227)
## Note: a particular data type could be selected. For D. melanogaster, Bgee has integarted 
## Affymetrix, RNA-seq, EST and in situ hybridization data
## Look at the data:
lapply(myTopAnatData, head)

## To perform the anatomical ontology enrichment test, you can readily 
## use the same gene list as used previously for topGO test
myTopAnatObject <-  topAnat(myTopAnatData, geneList)
## Warning: This step and the following take longer than the equivalent steps in topGO. This is 
## because the anatomical ontology is bigger than the Gene Ontology. Consider running a script 
## in batch mode if you have multiple analyses to perform.

## run the test
resultsAnatomy <- runTest(myTopAnatObject, algorithm = 'classic', statistic = 'fisher')

## Format the table of results, only displaying results significant at a 10% FDR threshold
myTableAnatomy <- makeTable(myTopAnatData, myTopAnatObject, resultsAnatomy, 0.1)

Question
What are the anatomical structures enriched for expression of DE genes? How does it relate to the GO enrichment results?

In population genomics studies, a major aim is often to demonstrate that a difference across individuals or across populations evolved under the action of positive selection, and is likely involved in some adaption. This is difficult to do with differential expression results because gene expression is a continuous character: it is hard to formulate a neutral model of evolution to use a null hypothesis. Many expression changes are likely to be neutral. Worse, it is difficult to pinpoint the precise genes onw hich selection could have acted since numerous expression changes could occur on genes that are regulated downstream of the causal/affected genes.

Thus it is nice to connect patterns of differential expression to patterns of sequence evolution. The analysis of sequence substitutions is a mature field, and there are numerous measures to detect positive selection on sequences. A significant expression change of a gene whose sequence experienced positive selection is a conspicuous evidence for the biological signficance of this gene.

Regarding the dataset you analyzed for this practical, have a look at the popDrowser http://popdrowser.uab.cat/, which provides a wide range of diversity/variation/selection measures across the D. melanogaster genome, calculated using the same DGRP lines. Many of these measures are very similar to the slots of PopGenome object you have created in the population genomics practicals earlier this week.

Question
Look at the different tracks available on this website. Are you familiar with these measures? Which ones would be most appropriate to search for positive selection on coding sequences across DGRP lines? What about regulatory sequences nearby genes?

If you are interested by a particular gene, you can easily visualize the sequence variation patterns in and around this genes. This could be useful to illustrate a talk or for a paper figure. If you want to download the preprocessed data across the whole genome, this seems possible, but the website is not very user friendly (after writing to the authors, they told me that an improved website, including more individuals from different populations and new population genetics estimates is in preparation and should be released this summer). I have found two main ways:

Question
Since this is a gene-based measure, it could be interesting to compare the neutrality index of DE genes vs. non-DE genes.

To do
If you have some time left, try playing around these measures and link them to your DE results. Good luck (and congrats if you went that far)!


Icons taken from http://www.flaticon.com/

Thanks to Amina Echchiki for proofreading and testing


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