vignettes/bifurcation_point.md

In hms-dbmi/crestree: Analysis of branching branching trajectories from single-cell data

Analysis of bifurcation points

This vignette describes analysis of individual bifurcation points based on a reconstructed transcriptional tree. As example, it explores bifurcation point between sensory and autonomic nervous systems in neural crest. The guideline starts with tree reconstruction, identifies fate-specific genes and estimates timing of their activation, assess existence and formation of fate-biases, and predicts time of genes inclusion in fate-biased phase.

Preliminaries: loading the libraries and neural crest data

library(igraph)
library(mgcv)
library(quadprog) 
library(pcaMethods) 
library(Rcpp) 
library(inline) 
library(RcppArmadillo) 
library(pbapply)
library(crestree)
library(ggplot2); library(gridExtra); library(grid);

data(crest)
emb <- crest$emb
clcol <- crest$clcol
nc.cells <- crest$nc.cells
wgm <- crest$wgm
wgwm <- crest$wgwm # matrix of expression weights
fpm <- read.table("http://pklab.med.harvard.edu/ruslan/neural_crest/fpm.txt",header=TRUE)
fpm <- as.matrix(fpm)
genes.tree <- crest$genes.tree

Run tree reconstruction

See manual the guide for tree reconstruction for detailed instructions. The tree can be either manually reconstructed:

M <- length(nc.cells); 
lambda <- 250; 
sigma <- 0.04
ppt <- ppt.tree(X=wgm[,nc.cells], W=wgwm[,nc.cells], emb=emb, lambda=250, sigma=0.04, metrics="cosine", M=M, err.cut = 5e-3, n.steps=30, seed=1, plot=FALSE)
ppt <- cleanup.branches(ppt,tips.remove = c(139,295))
ppt <- setroot(ppt,355)
ppt <- project.cells.onto.ppt(ppt,n.mapping = 100)

Alternatively, the tree object used in the paper can be downloaded from:

ppt <- readRDS(url("http://pklab.med.harvard.edu/ruslan/neural_crest/tree_structure_full.rds"))

Identification of branch-specific genes

Bifurcation point is charactarized by a progenitor and derivative branches.

plotppt(ppt,emb,tips=TRUE,forks=FALSE,cex.tree = 0.2,lwd.tree = 2)

We thus start with selection a root of progenitor branch (355) and two leaves of derivative branches (165 and 91):

root <- 355
leaves <- c(165,91)

Here is a fork charactarizing sensory-autonomic bifurcation point:

subtree <- extract.subtree(ppt,c(root,leaves))
plotppt(ppt,emb,tips=TRUE,forks=FALSE,cex.tree = 0.3,lwd.tree = 3,subtree=subtree)

A routine test.fork.genes performs assessment of genes differentially expression between post-bifurcation branches:

fork.de <- test.fork.genes(ppt,fpm,root=root,leaves=leaves,n.mapping = 10)

A table fork.de contains summary statistics of fold change effect, p-value p and adjusted p-value fdr of differential expression between branches, magnitude pd1.a (pd2.a) and p-value pd1.p (pd2.p) of expression changes from first (second) derivative branch to progenitor branch:

head(fork.de[order(fork.de$p),],)

| | effect| p| fdr| st| stf| pd1.a| pd1.p| pd2.a| pd2.p| |:-------|---------:|--:|---:|---:|---:|---------:|-----:|----------:|---------:| |Neurog2 | 1.9032796| 0| 0| 0.9| 0.9| 0.1012255| 0| -0.0262510| 0.0001858| |Dll1 | 1.4404789| 0| 0| 1.0| 1.0| 0.0976255| 0| 0.0043581| 0.3941055| |Srrm4 | 0.8383684| 0| 0| 1.0| 1.0| 0.0646809| 0| 0.0099734| 0.0002647| |Mfng | 0.9532576| 0| 0| 1.0| 1.0| 0.0630761| 0| 0.0034360| 0.4424417| |Eya2 | 1.0311280| 0| 0| 0.9| 0.9| 0.0737302| 0| 0.0082446| 0.0035890| |Pcdh8 | 1.1417276| 0| 0| 1.0| 0.9| 0.0515220| 0| -0.0193619| 0.0000199|

We next consider a gene to be preferentially expressed along the first (second) branch if it has effect.b1 (effect.b2) increased expression compared to another post-bifurcation branch and significant increase (p < 0.05) relative to progenitor branch:

fork.de <- branch.specific.genes(fork.de,effect.b1 = 0.1,effect.b2 = 0.3)

Column state characterizes genes that are specific to first (1), second (2), or neither (0) of derivative branches.

genes.sensory <- rownames(fork.de)[fork.de$state==1]
genes.autonomic  <- rownames(fork.de)[fork.de$state==2]

For consistency with the original results, we also limit genes to genes.tree set associated with the tree:

genes.sensory <- intersect(genes.sensory,genes.tree)
str(genes.sensory)
##  chr [1:98] "Rdh10" "Hes6" "Cxcr4" "Nfasc" "5730559C18Rik" "Rgs16" "Nhlh1" "Zbtb18" "Utrn" "Gamt" "Neurod4" "Upp1" ...

genes.autonomic <- intersect(genes.autonomic,genes.tree)
str(genes.autonomic)
##  chr [1:122] "Serpine2" "Lrrfip1" "Cdh19" "Ralb" "Angptl1" "Pbx1" "Mpz" "Lama4" "Cnn2" "Timp3" "Ascl1" "Elk3" ...

Classification of early and late modules

Dynamics of gene expression is reflected in timing of activation and expression optimum. Routine activation.fork estimates timing of optimum expression of smoothed expression and activation point as a first passage of derivative through deriv.cutoff cutoff:

fork.de.act <- activation.fork(ppt,fork.de,fpm,root,leaves,deriv.cutoff = 0.015,n.mapping=10)

fork.de.act table provides additional columns optimum and activation for genes predicted to be differentially expressed (stat = 1 or 2).

Branch-specific sets of genes (genes.sensory and genes.autonomic) can now be partitioned in early and late genes based on time of expression activation. A logical solution is to orient early/late genes relative to bifurcation point. Timing of root, bifurcation point and leaves are

fork.pt(ppt,root,leaves)

| root| bifurcation| leave 1| leave 2| |----:|-----------:|--------:|--------:| | 0| 17.5719| 27.59971| 31.49521|

We use cutoff=16.0 on timing of activation to define early and late genes:

cutoff <- 16.0

genes.sensory.late <- genes.sensory[fork.de.act[genes.sensory,]$activation > cutoff]
genes.sensory.early <- setdiff(genes.sensory,genes.sensory.late)

genes.autonomic.late <- genes.autonomic[fork.de.act[genes.autonomic,]$activation > cutoff]
genes.autonomic.early <- setdiff(genes.autonomic,genes.autonomic.late)

Now we can check if early/late genes modules follow co-activation or mutually-exclusive patterns:

cells <- rownames(ppt$cell.summary)[ppt$cell.summary$seg %in% extract.subtree(ppt,c(root,leaves))$segs]
par(mfrow=c(1,2))
plot(t(programs[c(1,3),cells]),col=ppt$cell.summary[cells,]$color,pch=19,cex=0.5)
plot(t(programs[c(2,4),cells]),col=ppt$cell.summary[cells,]$color,pch=19,cex=0.5)

Coordination of fate biases

Co-activation of both fate-specific programs poses a question of when cell acquire bias in favor of one or another program. For that, we look for coordinated expression of each module inside more homogeneous subpopulations. First, bifurcation fork is partitioned in non-intersecting windows of wind cells:

freq <- slide.cells(ppt,root,leaves,wind=50)

Visualization of group of cells assigned to each non-intersecting window:

fig_cells <- fig.cells(emb,freq)
marrangeGrob( c(fig_cells),ncol=length(fig_cells),nrow=1,top=NA)

Windows can be also selected manually, below we follow selection used in the paper:

regions = list( list(7,151,200,1),list(7,101,151,1),list(7,51,100,1),list(7,1,50,1),list(list(6,5,1,2),1,50, -1),list(list(6,5,1,2),51,100, -1),list(5,1,50,1),list(1,1,50,1))

freq <- slide.cells(ppt,root,leaves,wind=50,regions=regions)
fig_cells <- fig.cells(emb,freq)
marrangeGrob( c(fig_cells),ncol=length(fig_cells),nrow=1,top=NA)

Next, routine slide.cors estimates average correlation of each early fate-specific gene with both modules (genes.sensory.early and genes.autonomic.early) in each window of cells:

cors <- slide.cors(freq,fpm,genes.sensory.early,genes.autonomic.early)

Now joint visualization enables tracking how genes of fate-specific modules coordinate expression during progression along pseudotime:

fig_cor <- fig.cors(cors,genes.sensory.early,genes.autonomic.early)
marrangeGrob( c(fig_cells,fig_cor),ncol=length(fig_cells),nrow=2,
              layout_matrix = matrix(seq_len(2*length(fig_cells)), nrow = 2, ncol = length(fig_cells),byrow=TRUE),top=NA)

To obtain more contrasted (and reproducible with the paper) view, a set of early genes could be further cleaned up by removing fate-specific genes having low correlation with its modules around bifurcation point:

corA <- cors[[5]][,1]
genesetA <- names(which(corA[genes.sensory.early] > 0.07))

corB <- cors[[5]][,2]
genesetB <- names(which(corB[genes.autonomic.early] > 0.07))

Re-estimation average window-specific correlations for cleaned up sets of genes genesetA and genesetB:

cors <- slide.cors(freq,fpm,genesetA,genesetB)
fig_cor <- fig.cors(cors,genesetA,genesetB)
marrangeGrob( c(fig_cells,fig_cor),ncol=length(fig_cells),nrow=2,
              layout_matrix = matrix(seq_len(2*length(fig_cells)), nrow = 2, ncol = length(fig_cells),byrow=TRUE),top=NA)

More generally, formal trends of local coordination of fate-specific modules along branching trajectories can be estimated using synchro routine:

w=30
step=10
crd <- synchro(ppt,fpm,root,leaves,genesetA,genesetB,w,step,n.mapping=100,n.points = 300,span.smooth = 0.1,perm=FALSE)

And visualized:

visualize.synchro(crd)

Formation of correlated module

In order to infer genes orchestrating emergence of local intra-module correlations, we developed an iterative approach that estimates timing of genes inclusion in a core of correlated genes of the module. Routine onset estimates inclusion timing of early sensory genes genes.sensory.early for vector mappings of probabilistic cell projections:

inclusion.sensory <- onset(ppt,geneset=genes.sensory.early,nodes=c(root,leaves),expr=fpm,alp=20,w=40,step=10,n.cells=280,mappings=1:100,do.perm=FALSE,winperm=30,permut.n=10)

Average inclusion timing of each gene is

head(apply(inclusion.sensory,1,mean))
##         Rdh10          Hes6         Cxcr4 5730559C18Rik          Utrn          Gamt 
##      11.08199      12.97442      15.67575      12.01563      13.08833      15.35606

As an example, predicted cumulative probability of gene Pou4f1 inclusion is shown below:

show.gene.inclusion("Pou4f1",inclusion.sensory)

Overall summary statistics of inclusion probabilistics is visualized using show.inclusion.summary routine with each row of the heatmap reflecting cumulative inclusion probability of a single gene (as in example above):

show.inclusion.summary(inclusion.sensory,gns.show=c("Neurog2","Pou4f1"))

Analogous inference of inclusion timing for early atuonomic module genes.autonomic.early:

inclusion.autonomic <- onset(ppt,geneset=genes.autonomic.early,nodes=c(root,leaves),expr=fpm,alp=20,w=40,step=10,n.cells=280,mappings=1:100,do.perm=FALSE,winperm=30,permut.n=10)

show.inclusion.summary(inclusion.autonomic,gns.show = c("Mef2c","Pbx1"))

Finding optimal parameter to regulate false positive

For comparison, inclusion events are not detected in a control expression matrix, where expression levels were locally permuted using option do.perm=TRUE among winperm=10 batches of cells along pseudotime:

inclusion.sensory.control <- onset(ppt,geneset=genes.sensory.early,nodes=c(root,leaves),expr=fpm,alp=20,w=40,step=10,n.cells=280,
                 mappings=1:10,do.perm=TRUE,winperm=10,permut.n=10)
show.inclusion.summary(inclusion.sensory.control,gns.show=NA)

Parameter alp in routine onset, which was set to 20 in the analysis above, regulates stringency of inclusion point detection. Assessing summary statistics on inclusion times for a range of alp levels for real and permutated expression profiles enables selection of optimal alp. First, select a range of alp levels:

alps <- seq(1,40,length.out = 10)

Then estimate mean gene inclusion time for alps levels:

incl.real <- unlist(lapply(alps,function(alp){
  cat(alp);cat('\n')
  res.est <- onset(ppt,geneset=genes.sensory.early,nodes=c(root,leaves),expr=fpm,alp=alp,w=40,step=10,n.cells=280, mappings=1:10,do.perm=FALSE,winperm=10,permut.n=10)
  return(mean(res.est))
}))

Output is average inclusion time for each alp level:

head(incl.real)
## [1]  8.137490  9.916677 11.244118 11.996381 12.592830 13.203767

Estimate inclusion time for locally permuted expression levels by setting do.perm=TRUE and local permutation window winperm=10:

incl.perm <- unlist(lapply( alps,function(alp){
  cat(alp);cat('\n')
  res.est <- onset(ppt,geneset=genes.sensory.early,nodes=c(root,leaves),expr=fpm,alp=alp,w=40,step=10,n.cells=280, mappings=1:10,do.perm=TRUE,winperm=30,permut.n=10)
  mean(res.est)
}))

Comparison of early inclusion timing among all genes for real and control expression levels:

inclusion.stat(alps, incl.real, incl.perm)

From that one can see that if alp > 10 than overall false positive rate in permutations is close to zero, while detection sensitivity of real data gradually declines.

hms-dbmi/crestree documentation built on July 11, 2019, 8:04 p.m.