-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathset_analysis.Rmd
577 lines (430 loc) · 18.7 KB
/
set_analysis.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
---
title: "Characterising newly made gene sets"
author: "The DEE2 Gene Signatures Group"
date: "`r Sys.Date()`"
output:
html_document
---
Source: https://github.com/markziemann/dee2_gene_signatures
## Background
In this analysis we will be characterising the new gene sets.
## Libraries
```{r, libs}
suppressPackageStartupMessages({
library("getDEE2")
library("mitch")
library("triwise")
library("dplyr")
library("gplots")
library("reshape2")
library("network")
library("eulerr")
})
```
## Reference gene list
This is the background - all genes in the "universe" according to Ensembl version 90, which is the same annotation set used by DEE2.
You can see that there are a large number of non-protein coding genes.
```{r,genelist}
# get universe of gene names and biotypes
if (! file.exists("hs.gtf.gz") ) {
download.file("ftp://ftp.ensembl.org/pub/release-90/gtf/homo_sapiens/Homo_sapiens.GRCh38.90.gtf.gz",destfile= "hs.gtf.gz")
}
g<-read.table("hs.gtf.gz",sep="\t")
g <- g[grep("gene",g$V3),9]
universe <- sapply(strsplit(g," "),"[[",6)
universe <- gsub(";","",universe)
biotypes <- sapply(strsplit(g," "),"[[",10)
biotypes <- gsub("_"," ",biotypes)
biotypes <- gsub(";","",biotypes)
biotypes_df <- data.frame(universe,biotypes)
universe <- unique(universe)
mytable <- table(biotypes)
mytable <- mytable[order(mytable)]
mytable <- mytable[which(mytable>100)]
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.6,cex.axis = 0.6,
main="number of genes in each biotype class",xlim = c(0,20000)) ;grid()
mytable <- mytable/sum(mytable)*100
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.6,cex.axis = 0.6,
main="proportion (%) of genes in each biotype class" ,xlim=c(0,40)) ;grid()
# Take note how many transcripts are protein coding versus non-protein coding
paste("protein coding:",length(which(biotypes=="protein coding")))
paste("non-protein coding:",length(which(biotypes!="protein coding")))
```
## Curated gene sets
### Reactome
Here I download the current Reactome gene set library.
I would like to see how many Ensembl genes have some sort of annotated function.
Breaking it down into protein coding and non-protein coding, we can see that non-protein coding genes are severely underrepresented in Reactome.
Reactome has 2400 sets and a total of 11193 genes as of 29/Sep/2020.
```{r,reactome}
#o
download.file("https://reactome.org/download/current/ReactomePathways.gmt.zip", destfile="ReactomePathways.gmt.zip")
#unzip("ReactomePathways.gmt.zip")
reactome <- gmt_import("ReactomePathways.gmt")
reactome_genes <- unique(unname(unlist(reactome)))
length(reactome)
length(reactome_genes)
v1 <- list("Reactome"=reactome_genes, "Ensembl universe"=universe)
library("eulerr")
plot(euler(v1),quantities = TRUE)
length(reactome_genes)/length(universe)*100
prot <- biotypes_df[which(biotypes_df$biotypes=="protein coding"),1]
prot <- unique(prot)
nprot <- biotypes_df[which(biotypes_df$biotypes!="protein coding"),1]
nprot <- unique(nprot)
intersect(prot,nprot)
nprot <- setdiff(nprot,prot)
v1 <- list("Reactome"=reactome_genes, "protein coding"=prot, "non-protein coding"=nprot)
plot(euler(v1),quantities = TRUE)
```
### Gene Ontology
Now let's do the same with GO sets downloaded from Ensembl biomart.
GO has 12264 sets and 8412 genes as of 26/Sep/2020.
```{r,go}
go <- read.table("biomart_2020-09-26.txt.gz",sep="\t", fill = TRUE)
colnames(go) <- go[1,]
go <- go[2:nrow(go),]
go <- go[which(go$`GO term accession`!=""),]
go_genes <- unique(go$`Gene name`)
length(unique(go$`GO term accession`))
length(go_genes)
v1 <- list("GO"=go_genes, "protein coding"=prot, "non-protein coding"=nprot)
plot(euler(v1),quantities = TRUE)
length(unique(go$`GO term accession`))
head(go)
```
### Human Phenotype Ontology
Now same for Human phenotype ontology.
Downloaded from https://hpo.jax.org/app/download/annotation.
There are 2958 sets and 4366 genes as of 27/Sep/2020.
```{r,hpo}
hpo <- read.table("phenotype_to_genes.txt",sep="\t")
hpo_genes <- unique(hpo$V4)
head(hpo)
length(unique(hpo$V1))
length(hpo_genes)
v1 <- list("HPO"=hpo_genes, "protein coding"=prot, "non-protein coding"=nprot)
plot(euler(v1),quantities = TRUE)
```
### MSigDB
Now for MSigDB version 7.2 accessed 27/Sep/2020.
There are 31120 sets and 40044 genes.
```{r,msigdb}
msigdb <- gmt_import("msigdb.v7.2.symbols.gmt")
msigdb_genes <- unique(unname(unlist(msigdb)))
length(msigdb)
length(msigdb_genes)
v1 <- list("MSigDB"=msigdb_genes,"protein coding"=prot, "non-protein coding"=nprot)
plot(euler(v1),quantities = TRUE)
```
## Defining the overrepresentation analysis function
Here I'm using the triwise::testEnrichment function which uses a Fisher test under the hood.
Reference
van de Laar L, Saelens W, De Prijck S, Martens L, Scott CL, Van Isterdael G, Hoffmann E, Beyaert R, Saeys Y, Lambrecht BN, Guilliams M. Yolk Sac Macrophages, Fetal Liver, and Adult Monocytes Can Colonize an Empty Niche and Develop into Functional Tissue-Resident Macrophages. Immunity. 2016 Apr 19;44(4):755-68. doi: 10.1016/j.immuni.2016.02.017. Epub 2016 Mar 15. PMID: 26992565.
```{r,orafunc}
ora <- function(gs,genesets,universe){
res <- triwise::testEnrichment(gs, genesets, universe, minknown = 3, mindiffexp = 2, maxknown = 5000)
if( nrow(res)>0 ) {
res <- res[order(res$pval),]
res <- head(res,5)
hits <- log(res[which(res$qval<0.01),2])
names(hits) <- res[which(res$qval<0.01),4]
return(hits)
} else {
return(0)
}
}
```
Here is a function that converts list of vectors to a network diagram.
There are 2x edges than nodes.
Only the edges with highest similarity are retained, as per jaccard.
The size of the gene set is proportional to the node size (sqrt).
```{r,gsnet}
gs2net <- function(gset){
mydf <- bind_rows(lapply(gset, as.data.frame.list))
rownames(mydf) <- names(gset)
j <- apply(mydf,1,function(x) {
apply(mydf,1,function(y) {
length(intersect(x,y) ) / length(union(x,y))
})
})
j[lower.tri(j)] <- NA
j[lower.tri(j,diag=TRUE)] <- 0
jl <- melt(j)
jl <- jl[which(jl$Var1 != jl$Var2),]
jl <- jl[which(jl$value != 1),]
jl <- jl[order(-jl$value),]
jl <- head(jl,length(gset)*2)
jl$edgeSize = with(jl, jl$value/sum(jl$value))
lengths <- unlist(lapply(gset,length))
lengths <- sqrt(lengths/sum(lengths)*100)
jl$vertexsize <- lengths[match(as.character(jl$Var1),names(lengths))]
jlnet = network(jl[,1:2])
plot(jlnet, displaylabels = TRUE, label.col = "steelblue",
edge.lwd = c(jl$edgeSize) * 100,
arrowhead.cex = 0,
label.cex = 1, vertex.border = "white",vertex.cex = jl$vertexsize,
vertex.col = "blue", edge.col = rgb(0, 0, 0, alpha = 0.5))
}
```
## Epilepsy
Let's analyse the new epilepsy gene sets.
There are only 12 sets.
We begin with a histogram of the set size.
```{r,epilepsy1}
epi <- gmt_import("epilepsy_genesymbols.gmt")
epi <- diab[which(!names(epi)=='')]
epi <- epi[which(!duplicated(names(epi)))]
length(epi)
setnames <- names(epi)
names(epi) <-paste("GS",1:length(epi))
numgenes <- unlist(lapply(epi,length))
hist(numgenes,breaks = 15, xlab="number of genes per set",main="Epilepsy gene sets")
```
Next we can look at the similarity between the gene sets with a network diagram.
GS8, GS12 and GS10 have some overlap, as do GS7 with GS11.
Next we can see that the gene sets have diverse representation of biotypes.
There is a high proportion of protein coding genes but also lincRNA.
```{r,epilepsy2,fig.width=8,fig.height=6}
gs2net(epi)
# classes
mytable <- lapply(epi,function(x) {
table(biotypes_df[which(x %in% biotypes_df$universe),2])
})
mydf <- bind_rows(lapply(mytable, as.data.frame.list))
rownames(mydf) <- names(epi)
mydf_n <- mydf/rowSums(mydf)
mydf <- mydf[,head(order(-colSums(mydf_n)),8)]
colfunc <- colorRampPalette(c("white", "yellow","orange","red","darkred"))(n=25)
heatmap.2(t(as.matrix(mydf)),col=colfunc,trace="none",margin=c(5,10),cexCol = 0.9, cexRow = 0.9,
scale="col",dendrogram = "none",main="Epilepsy gene sets by biotype",na.color = "white")
```
The overall biotype has a high representaiton of protein coding genes similar to Reactome.
We also observed a higher representation of lincRNAs in these gene sets.
Pseudogenes were underrepresented.
```{r,epilepsy3}
mydf_n[is.na(mydf_n)] <- 0
mytable <-colSums(mydf_n)
mytable <- mytable/sum(mytable)*100
mytable<-mytable[order(mytable)]
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.7,cex.axis = 0.8,
main="Overall biotype class representation (%)" ) ;grid()
```
Next we wanted to know the Reactome functional categories that are enriched in each of the gene sets.
Using a Fisher text, we performed an enrichment analysis with Reactome genes using Ensembl as the background (universe).
We are aware that it may the improper background and the best background would be all genes that were detected in the original experiment.
As expected, some of these sets are related to neurological functions.
```{r,epilepsy4,fig.width=10,fig.height=8}
# ORA
res <- lapply(epi, function(x) { ora(gs=x,genesets=reactome, universe=universe) } )
res <- res[which(lapply(res,length)>0)]
myres <- bind_rows(lapply(res, as.data.frame.list))
myres[is.na(myres)] <- 0
rownames(myres) <- names(res)
colfunc <- colorRampPalette(c("white", "lightblue","blue", "darkblue"))(n=25)
heatmap.2(t(as.matrix(myres)),col=colfunc,trace="none",margin=c(10,25),cexRow = 0.6,
dendrogram = "none",main="Epilepsy gene sets by Reactome enrichment")
```
## Diabetes
Now lets analyse the new diabetes gene sets.
There are 129 gene sets.
```{r,diabetes1}
diab <- gmt_import("diabetes_genesymbols.gmt")
diab <- diab[which(!names(diab)=='')]
diab <- diab[which(!duplicated(names(diab)))]
length(diab)
setnames <- names(diab)
names(diab) <-paste("GS",1:length(diab))
numgenes <- unlist(lapply(diab,length))
hist(numgenes,breaks = 15, xlab="number of genes per set",main="Diabetes gene sets")
```
The network diagram indicates clustering into a few groups.
A large group with GS39 at the centre (right side).
A smaller cluster around GS121 (left side).
A smaller cluster around GS120 (top).
The biotype heatmap indicates representation of protein coding, lincRNAs and processed pseudgenes.
In some sets, non-coding genes outnumber protein coding genes (left).
In most sets, protein coding genes are more abundant (right).
```{r,diabetes2,fig.width=8,fig.height=6}
gs2net(diab)
# classes
mytable <- lapply(diab,function(x) {
table(biotypes_df[which(x %in% biotypes_df$universe),2])
})
mydf <- bind_rows(lapply(mytable, as.data.frame.list))
rownames(mydf) <- names(diab)
mydf_n <- mydf/rowSums(mydf)
mydf <- mydf[,head(order(-colSums(mydf_n)),8)]
colfunc <- colorRampPalette(c("white", "yellow","orange","red","darkred"))(n=25)
heatmap.2(t(as.matrix(mydf)),col=colfunc,trace="none",margin=c(5,10),cexCol = 0.3,
cexRow = 0.6 ,scale="col",dendrogram = "none",main="Diabetes gene sets by biotype",na.color = "white")
```
In these gene sets, psudogenes are underrepresented while protein coding, lincRNA and antisense RNA genes are overrepresented.
```{r,diabetes3}
mydf_n[is.na(mydf_n)] <- 0
mytable <-colSums(mydf_n)
mytable <- mytable/sum(mytable)*100
mytable<-mytable[order(mytable)]
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.7,cex.axis = 0.8,
main="Overall biotype class representation (%)" ) ;grid()
```
Here is the reactome enrichment plot for the diabetes gene sets.
The text is too small to make out from this chart so I will need to look at making a PDF version.
One thing we can see from the chart is that not many of the enriched Reactomes are common apart from the horizontal strip near the bottom.
```{r,diabetes4,fig.width=10,fig.height=8}
# reactome enrichment
res <- lapply(diab, function(x) { ora(gs=x,genesets=reactome, universe=universe) } )
res <- res[which(lapply(res,length)>0)]
myres <- bind_rows(lapply(res, as.data.frame.list))
myres[is.na(myres)] <- 0
rownames(myres) <- names(res)
colfunc <- colorRampPalette(c("white", "lightblue","blue", "darkblue"))(n=25)
heatmap.2(t(as.matrix(myres)),col=colfunc,trace="none",margin=c(5,10),cexRow = 0.2,
cexCol = 0.3 ,dendrogram = "none",main="Diabetes gene sets by Reactome enrichment")
```
## Heart disease
Let's analyse the new heart disease gene sets.
There are 50 gene sets.
```{r,heartdisease1}
hd <- gmt_import("heartdisease_genesymbols.gmt")
hd <- diab[which(!names(hd)=='')]
hd <- hd[which(!duplicated(names(hd)))]
length(hd)
setnames <- names(hd)
names(hd) <-paste("GS",1:length(hd))
numgenes <- unlist(lapply(hd,length))
hist(numgenes,breaks = 15, xlab="number of genes per set",main="Heart disease gene sets")
```
The network chart indicatesmost sets are linked around GS50.
There are several very small groups on the right of the chart.
The biotype heatmap shows overall genes sets are composed motly of protein coding genes and a small number of sets have a majority of lincRNA and pseudo genes.
```{r,heartdisease2,fig.width=8,fig.height=6}
gs2net(hd)
# classes
mytable <- lapply(hd,function(x) {
table(biotypes_df[which(x %in% biotypes_df$universe),2])
})
mydf <- bind_rows(lapply(mytable, as.data.frame.list))
rownames(mydf) <- names(hd)
mydf_n <- mydf/rowSums(mydf)
mydf <- mydf[,head(order(-colSums(mydf_n)),8)]
colfunc <- colorRampPalette(c("white", "yellow","orange","red","darkred"))(n=25)
heatmap.2(t(as.matrix(mydf)),col=colfunc,trace="none",margin=c(5,10),cexCol = 0.2,
cexRow = 0.8,scale="col",dendrogram = "none",main="Heart disease gene sets by biotype",na.color = "white")
```
Similar to the other sets, pseudogenes are underrepresented and protein coding, lincRNA, and antisense genes are overrepresented.
```{r,heartdisease3}
mydf_n[is.na(mydf_n)] <- 0
mytable <-colSums(mydf_n)
mytable <- mytable/sum(mytable)*100
mytable<-mytable[order(mytable)]
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.7,cex.axis = 0.8,
main="Overall biotype class representation (%)" ) ;grid()
```
The reactome heatmap shows a cluster gene sets that are enriched for ECM, collagen and IGF signaling (lower).
There is a strip showing metabolism, immune signaling and signal transduction are common.
Apart from that, the enriched gene sets are quite distinct.
```{r,heartdisease4,fig.width=10,fig.height=8}
# reactome enrichment
res <- lapply(hd, function(x) { ora(gs=x,genesets=reactome, universe=universe) } )
res <- res[which(lapply(res,length)>0)]
myres <- bind_rows(lapply(res, as.data.frame.list))
myres[is.na(myres)] <- 0
rownames(myres) <- names(res)
colfunc <- colorRampPalette(c("white", "lightblue","blue", "darkblue"))(n=25)
heatmap.2(t(as.matrix(myres)),col=colfunc,trace="none",margin=c(5,15),cexRow = 0.4, cexCol = 0.5,
dendrogram = "none",main="Heart disease gene sets by Reactome enrichment")
```
## SARS, MERS and SARS-CoV-2
Let's analyse the new virus infection gene sets.
There are 66 sets.
```{r,sars1}
sars <- gmt_import("sarsmers_genesymbols.gmt")
sars <- diab[which(!names(sars)=='')]
sars <- sars[which(!duplicated(names(sars)))]
length(sars)
setnames <- names(sars)
names(sars) <-paste("GS",1:length(sars))
numgenes <- unlist(lapply(sars,length))
hist(numgenes,breaks = 15, xlab="number of genes per set",main="SARS, MERS and SARS-CoV-2 gene sets")
```
The network diagram shows a large cluster of sets around GS21 (left side).
A smaller cluster exists around GS24 (right side) and GSGS29 (bottom).
The biotype chart shows most gene sets are dominated with protein coding genes but about 1/3 have dominance of lincRNA and pseudogenes.
```{r,sars2,fig.width=8,fig.height=6}
gs2net(sars)
# classes
mytable <- lapply(sars,function(x) {
table(biotypes_df[which(x %in% biotypes_df$universe),2])
})
mydf <- bind_rows(lapply(mytable, as.data.frame.list))
rownames(mydf) <- names(sars)
mydf_n <- mydf/rowSums(mydf)
mydf <- mydf[,head(order(-colSums(mydf_n)),8)]
colfunc <- colorRampPalette(c("white", "yellow","orange","red","darkred"))(n=25)
heatmap.2(t(as.matrix(mydf)),col=colfunc,trace="none",margin=c(5,10),cexCol = 0.4, cexRow = 0.8,
scale="col",dendrogram = "none",main="SARS, MERS and SARS-CoV2 gene sets by biotype",na.color = "white")
```
The overall biotype representation is similar to the other sets with overrepresentation of lincRNA and antisense RNA genes.
```{r,sars3}
mydf_n[is.na(mydf_n)] <- 0
mytable <-colSums(mydf_n)
mytable <- mytable/sum(mytable)*100
mytable<-mytable[order(mytable)]
par(mar=c(3,13,1,1)); barplot(mytable,horiz=TRUE,las=2,cex.names = 0.7,cex.axis = 0.8,
main="Overall biotype class representation (%)" ) ;grid()
```
There is a cluster of immune related enrichment in several gene sets as expected (lower left).
Moreover there are some clustering of integrins (lower left), and cell cycle (middle).
Post translational modification and signal transduction were also common.
```{r,sars4,fig.width=10,fig.height=8}
# Reactome enrichment
res <- lapply(sars, function(x) { ora(gs=x,genesets=reactome, universe=universe) } )
res <- res[which(lapply(res,length)>0)]
myres <- bind_rows(lapply(res, as.data.frame.list))
myres[is.na(myres)] <- 0
rownames(myres) <- names(res)
colfunc <- colorRampPalette(c("white", "lightblue","blue", "darkblue"))(n=25)
heatmap.2(t(as.matrix(myres)),col=colfunc,trace="none",margin=c(5,10),cexRow = 0.3, cexCol = 0.4,
dendrogram = "none",main="SARS, MERS and SARS-CoV-2 gene sets genes by Reactome enrichment")
```
## Overall analysis
There are 257 gene sets in total.
The median set size was smallest for epilepsy and largest for diabetes.
In terms of number of genes in each library, epilepsy has 2175, diabetes has 11179, heart disease has 8727 and virus has 6827.
There were 731 genes that were common to all three libraries.
Altogether these new sets include 15522 genes, of these 12824 are protein coding and 2698 are non-protein coding.
```{r,overall1}
names(epi) <- paste("Epilepsy",names(epi) )
names(diab) <- paste("Diabetes",names(diab) )
names(hd) <- paste("Heart disease",names(hd) )
names(sars) <- paste("Virus related",names(sars) )
all <- c(epi,diab,hd,sars)
length(all)
barplot( c(length(epi) , length(diab) , length(hd) , length(sars) ) ,
names.arg = c("Epilepsy","Diabetes","Heart Disease","Virus") ,
ylab= "number of gene sets")
epil <- sapply(epi,length)
diabl <- sapply(diab,length)
hdl <- sapply(hd,length)
sarsl <- sapply(sars,length)
boxplot(epil,diabl,hdl,sarsl,names=c("Epilepsy","Diabetes","Heart Disease","Virus"))
epig <- unique(unname(unlist(epi)))
diabg <- unique(unname(unlist(diab)))
hdg <- unique(unname(unlist(hd)))
sarsg <- unique(unname(unlist(sars)))
length(epig)
length(diabg)
length(hdg)
length(sarsg)
v1 <- list("Epilepsy"=epig, "Diabetes"=diabg, "Heart diabetes"=hdg, "Virus related" = sarsg)
plot(euler(v1),quantities = TRUE)
allg <- unique(unname(unlist(all)))
length(allg)
v1 <- list("DEE2 sets"=allg,"protein coding"=prot, "non-protein coding"=nprot)
plot(euler(v1),quantities = TRUE)
```
## Session information
```{r sessioninfo}
sessionInfo()
```