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.

Since this is an anlaysis in progress, there is a lot we do not know about these data.

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  11

Here 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")
}

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.

library(genlasso)

Loading required package: MASS

Loading required package: Matrix

Loading required package: igraph

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union

d2.tf = trendfilter(d[,2],ord = 1)

Warning: 'rBind' is deprecated.
 Since R version 3.2.0, base's rbind() should work fine with S4 objects

d2.tf.cv = cv.trendfilter(d2.tf) # performs 5-fold CV

Fold 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 CV

Fold 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 CV

Fold 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)

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 CV

Fold 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)

Wavelets

Here we will apply wavelets to smooth these data.

To apply wavelets 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.

library("wavethresh")

WaveThresh: R wavelet software, release 4.6.8, installed

Copyright Guy Nason and others 1993-2016

Note: nlevels has been renamed to nlevelsWT

wds <- wd(d.sub[,2],family="DaubExPhase",filter.number = 1)
plot(wds)

[1] 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009

To 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] 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009 12.1009

Now 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)

What happens if we use an Empirical Bayes thresholding rule?

The EbayesThresh package essentially solves the EBNM problem, using a prior distribution that is a mixture of a point mass at 0 and a Laplace distribution with rate parameter \(a\): \[ \pi_0 \delta_0 + (1-\pi_0) DExp(a).\]

Here a=NA tells Ebayesthresh ot estimate \(a\). Notice that the outliers cause “problems”

library("EbayesThresh")
wds <- wd(d.sub[,2],family="DaubLeAsymm",filter.number = 8)
wtd <- ebayesthresh.wavelet(wds,a=NA)
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)

Try removing the outliers (actually setting them to the mean here), just to see what happens.

xx = ifelse(d.sub[,2]<2,mean(d.sub[,2]),d.sub[,2])
wds <- wd(xx,family="DaubLeAsymm",filter.number = 8)
wtd <- ebayesthresh.wavelet(wds,a=NA)
fd <- wr(wtd) #reconstruct
plot(d.sub$theta,xx,xlab="cell cycle", ylab = "Expression")
lines(d.sub$theta,fd,col=2,lwd=3)

Notice that estimating \(a\) is really important (the default is to set \(a=0.5\)):

xx = ifelse(d.sub[,2]<2,mean(d.sub[,2]),d.sub[,2])
wds <- wd(xx,family="DaubLeAsymm",filter.number = 8)
wtd <- ebayesthresh.wavelet(wds)
fd <- wr(wtd) #reconstruct
plot(d.sub$theta,xx,xlab="cell cycle", ylab = "Expression")
lines(d.sub$theta,fd,col=2,lwd=3)

sessionInfo()

R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

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] EbayesThresh_1.5-3 wavethresh_4.6.8   genlasso_1.3      
[4] igraph_1.2.1       Matrix_1.2-14      MASS_7.3-49       

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.0.1   pscl_1.5.2       
 [7] doParallel_1.0.11 SQUAREM_2017.10-1 lattice_0.20-35  
[10] foreach_1.4.4     ashr_2.2-7        stringr_1.3.0    
[13] tools_3.3.2       parallel_3.3.2    grid_3.3.2       
[16] R.oo_1.22.0       git2r_0.21.0      iterators_1.0.9  
[19] htmltools_0.3.6   yaml_2.1.18       rprojroot_1.3-2  
[22] digest_0.6.15     codetools_0.2-15  R.utils_2.6.0    
[25] evaluate_0.10.1   rmarkdown_1.9     stringi_1.1.7    
[28] backports_1.1.2   R.methodsS3_1.7.1 truncnorm_1.0-7  
[31] pkgconfig_2.0.1