Cell cycle genes
Matthew Stephens
April 9, 2018
Last updated: 2021-04-13
Checks: 7 0
Knit directory: stat34800/analysis/
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| File | Version | Author | Date | Message |
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| Rmd | ae40d4b | Matthew Stephens | 2021-04-13 | workflowr::wflow_publish(“cell_cycle.Rmd”) |
| html | c02c5c1 | Matthew Stephens | 2021-04-13 | Build site. |
| Rmd | f5b651b | Matthew Stephens | 2021-04-13 | workflowr::wflow_publish(“cell_cycle.Rmd”) |
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| Rmd | d63b9eb | stephens999 | 2018-05-21 | workflowr::wflow_publish(“svd_zip.Rmd”) |
| Rmd | 3d82c11 | stephens999 | 2018-04-11 | update cell cycle example to include circularity of trend filtering |
| Rmd | 463d7d9 | stephens999 | 2018-04-11 | update to include trend filter |
| Rmd | 8b80a72 | stephens999 | 2018-04-09 | add cell cycle example |
Load Packages
You will need to have genlasso, wavethresh and smashr installed to run. The first two are on CRAN. The smashr package can be installed from github using the following code:
devtools::install_github("stephenslab/smashr")library(wavethresh)Loading required package: MASSWarning: package 'MASS' was built under R version 3.6.2WaveThresh: R wavelet software, release 4.6.8, installedCopyright Guy Nason and others 1993-2016Note: nlevels has been renamed to nlevelsWTlibrary(genlasso)Warning: package 'genlasso' was built under R version 3.6.2Loading required package: MatrixLoading required package: igraph
Attaching package: 'igraph'The following objects are masked from 'package:stats':
decompose, spectrumThe following object is masked from 'package:base':
unionlibrary(smashr)Some Cell Cycle Data
The data come from a recent experiment performed in the Gilad lab by Po Tung, in collaboration with Joyce Hsiao and others. This was an early look at these data (later published in [https://genome.cshlp.org/content/30/4/611]).
The data are measuring the activity of 10 genes that may or may not be involved in the “cell cycle”, which is the process cells go through as they divide. (We have data on a large number of genes, but Joyce has picked out 10 of them for us to look at.) Each gene is measured in many single cells, and we have some independent (but noisy) measurement of where each cell is in the cell cycle.
d = readRDS("../data/cyclegenes.rds")
dim(d)[1] 990 11Here each row is a single cell. The first column (“theta”) is an estimate of where that cell is in the cell cycle, from 0 to 2pi. (Note that we don’t know what stage of the cell cycle each point in the interval corresponds to - so there is no guarantee that 0 is the “start” of the cell cycle. Also, because of the way these data were created we don’t know which direction the cell cycle is going - it could be forward or backward.) Then there are 10 columns corresponding to 10 different genes.
I’m going to order the rows by cell cycle (theta, first column) as this will make things much easier later.
# order the data
o = order(d[,1])
d = d[o,]
plot(d$theta)
Here we just plot 8 genes to get a sense for the data:
par(mfcol=c(2,2))
for(i in 1:4){
plot(d$theta, d[,(i+1)],pch=".",ylab="gene expression",xlab="cell cycle location")
}
par(mfcol=c(2,2))
for(i in 1:4){
plot(d$theta, d[,(i+5)],pch=".",ylab="gene expression",xlab="cell cycle location")
}
| Version | Author | Date |
|---|---|---|
| c02c5c1 | Matthew Stephens | 2021-04-13 |
The question we want to answer is this: which genes show greatest evidence for varying in their expression through the cell cycle? For now what we really want is a filter we can apply to a large number of genes to pick out the ones that look most “interesting”. Later we might want a more formal statistical measure of evidence.
Our current idea is to try “smoothing” these data, and pick out genes where the change in the mean over theta is most variable (in some sense). The extreme would be if the smoother fits a horizontal line - that indicates no variability with theta, so those genes are not interesting to us.
Trend filtering
Here we will apply trend filtering to smooth these data. Trend filtering, at its simplest, applies L1 regularization to the changes in mean from one observation to the next. (The extreme would be no changes in any of these means, so a flat line.) It is implemented in the “genlasso” package.
d2.tf = trendfilter(d[,2],ord = 1)
d2.tf.cv = cv.trendfilter(d2.tf) # performs 5-fold CVFold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... plot(d[,1],d[,2],xlab="cell cycle",ylab="expression")
lines(d[,1],predict(d2.tf, d2.tf.cv$lambda.min)$fit,col=2,lwd=3)
This fit is a bit on the “spiky” side in places. We can get a smoother fit by using a higher order filter. The details are too much to include here, but basically instead of shrinking first order differences it shrinks things that also measure differences with neighbors a bit further apart (2nd order).
d2.tf2 = trendfilter(d[,2],ord = 2)
d2.tf2.cv = cv.trendfilter(d2.tf2) # performs 5-fold CVFold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... plot(d[,1],d[,2],xlab="cell cycle",ylab="expression")
lines(d[,1],predict(d2.tf, d2.tf.cv$lambda.min)$fit,col=2,lwd=3)
lines(d[,1],predict(d2.tf2, d2.tf2.cv$lambda.min)$fit,col=3,lwd=3)
And here we try another gene that maybe shows less evidence for variability.
d7.tf2 = trendfilter(d[,7],ord = 2)
d7.tf2.cv = cv.trendfilter(d7.tf2) # performs 5-fold CVFold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... plot(d[,1],d[,7],xlab="cell cycle",ylab="expression")
lines(d[,1],predict(d7.tf2, d7.tf2.cv$lambda.min)$fit,col=3,lwd=3)
| Version | Author | Date |
|---|---|---|
| c02c5c1 | Matthew Stephens | 2021-04-13 |
Dealing with the circularity
Because the \(x\) axis here is cyclical, the value of \(E(Y|x)\) near \(x=0\) should be similar to the value near \(x=2pi\). But trend filtering does not know this. We can encourage this behaviour by duplicating the data using a translation. (Note this is different than reflecting it about the boundaries).
Here is an example:
yy = c(d[,2],d[,2],d[,2]) ## duplicated data
xx = c(d[,1]-2*pi, d[,1], d[,1]+2*pi) # shifted/translated x coordinates
yy.tf2 = trendfilter(yy,ord = 2)
yy.tf2.cv = cv.trendfilter(yy.tf2) # performs 5-fold CVFold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... plot(xx,yy,xlab="cell cycle",ylab="expression")
lines(xx,predict(yy.tf2, yy.tf2.cv$lambda.min)$fit,col=3,lwd=3)
# plot only a single version of data
include = c(rep(FALSE,length(d[,2])), rep(TRUE, length(d[,2])), rep(FALSE, length(d[,2])))
plot(xx[include],yy[include],xlab="cell cycle",ylab="expression", main="trend filtering with circular fit")
lines(xx[include],predict(yy.tf2, yy.tf2.cv$lambda.min)$fit[include],col=3,lwd=3)
| Version | Author | Date |
|---|---|---|
| c02c5c1 | Matthew Stephens | 2021-04-13 |
Wavelets
Here we will apply wavelets to smooth these data.
To apply basic wavelets methods we need the data to be a power of 2. Also we need the data to be ordered in terms of theta. We’ll subset the data to 512 elements here, and order it:
# subset the data
set.seed(1)
subset = sort(sample(1:nrow(d),512,replace=FALSE))
d.sub = d[subset,]Here we do the Haar wavelet by specifying family="DaubExPhase",filter.number = 1 to the discrete wavelet transform function wd. The plot shows the wavelet transformed values, separately at each resolution.
wds <- wd(d.sub[,2],family="DaubExPhase",filter.number = 1)
plot(wds)
[1] 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629
[9] 12.31629To illustrate the idea behind wavelet shrinkage we use the policy “manual” to shrink all the high-resolution coefficients (levels 4-8) to 0.
wtd <- threshold(wds, levels = 4:8, policy="manual",value = 99999)
plot(wtd) 
[1] 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629 12.31629
[9] 12.31629Now undo the wavelet transform on the shrunken coefficients
fd <- wr(wtd) #reconstruct
plot(d.sub$theta,d.sub[,2],xlab="cell cycle", ylab = "Expression")
lines(d.sub$theta,fd,col=2,lwd=3)
The estimate here is a bit “jumpy”, due to the use of the Haar wavelet and the rather naive hard thresholding. We can make it less “jumpy” by using a “less step-wise” wavelet basis
wds <- wd(d.sub[,2],family="DaubLeAsymm",filter.number = 8)
wtd <- threshold(wds, levels = 4:8, policy="manual",value = 99999)
fd <- wr(wtd) #reconstruct
plot(d.sub$theta,d.sub[,2],xlab="cell cycle", ylab = "Expression")
lines(d.sub$theta,fd,col=2,lwd=3)
Emprical Bayes approach to wavelet smoothing
The smashr package described here essentially does wavelet smoothing using an Empirical Bayes method to estimate the prior on the wavelet coefficients. By estimating the prior from the data smashr decides how much to shrink each wavelet coefficient. Also smashr does not require the data to be a power of 2, so we can apply it directly.
However, out of the box we can see the outliers cause “problems”.
smash.res = smash(d[,2])
plot(d$theta,d[,2],xlab="cell cycle", ylab = "Expression")
lines(d$theta,smash.res,col=2,lwd=3)
Here we removed the outliers (actually setting them to the mean) to avoid this problem. The resulting fit is much better.
xx = ifelse(d[,2]<2,mean(d[,2]),d[,2])
smash.res = smash(xx)
plot(d$theta,d[,2],xlab="cell cycle", ylab = "Expression")
lines(d$theta,smash.res,col=2,lwd=3)
| Version | Author | Date |
|---|---|---|
| c02c5c1 | Matthew Stephens | 2021-04-13 |
The above uses the default wavelet, which for smash is the Haar wavelet. The fit is less step-like than you might expect with the Haar wavelet both because the EB shrinkage produces a natural smoothing property (the posterior mean is not exactly sparse) and because smashr uses a rotation scheme to sum over all possibly rotations of the data (sometimes called the “Translation invariant” wavelet transform.) However we can change the default by specifying family and/or filter number. Here we see that changing filter.number to 8 does not make much difference:
smash.res.8 = smash(xx, filter.number=8)
plot(d$theta,d[,2],xlab="cell cycle", ylab = "Expression")
lines(d$theta,smash.res,col=2,lwd=3)
lines(d$theta,smash.res.8,col=3,lwd=3)
Dealing with the circularity
One could deal with circularity the same way as in trend-filtering.
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] smashr_1.2-7 genlasso_1.5 igraph_1.2.5 Matrix_1.2-18
[5] wavethresh_4.6.8 MASS_7.3-51.6
loaded via a namespace (and not attached):
[1] Rcpp_1.0.6 pillar_1.4.6 compiler_3.6.0 later_1.1.0.1
[5] git2r_0.27.1 workflowr_1.6.2 bitops_1.0-6 tools_3.6.0
[9] digest_0.6.27 evaluate_0.14 lifecycle_1.0.0 tibble_3.0.4
[13] lattice_0.20-41 pkgconfig_2.0.3 rlang_0.4.10 rstudioapi_0.13
[17] yaml_2.2.1 xfun_0.16 invgamma_1.1 stringr_1.4.0
[21] knitr_1.29 caTools_1.18.0 fs_1.5.0 vctrs_0.3.4
[25] rprojroot_1.3-2 grid_3.6.0 data.table_1.12.8 glue_1.4.2
[29] R6_2.4.1 rmarkdown_2.3 mixsqp_0.3-43 irlba_2.3.3
[33] ashr_2.2-51 magrittr_1.5 whisker_0.4 backports_1.1.10
[37] promises_1.1.1 ellipsis_0.3.1 htmltools_0.5.0 httpuv_1.5.4
[41] stringi_1.4.6 truncnorm_1.0-8 SQUAREM_2020.3 crayon_1.3.4