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Here we examine and compare the topic modeling results for the two closely related data sets from Zheng et al (2017), the mixture of FACS-purified PBMC data sets, and the “unsorted” 68k PBMC data. The goal of this analysis is to illustrate how the topic models fitted to these data sets can be used to learn about structure in the data. In particular, we would like to identify clusters, and interpret clusters and topics as “cell types” or “gene expression programs”.

Load the packages used in the analysis below, as well as additional functions that will be used to generate some of the plots.

library(dplyr)
library(fastTopics)
library(ggplot2)
library(cowplot)
source("../code/plots.R")

Mixture of FACS-purified PBMC data

We begin with the mixture of FACS-purified PBMC data. Note that the count data are no longer needed at this stage.

load("../data/pbmc_purified.RData")
samples_purified <- samples
rm(samples,genes,counts)

Load the \(k = 6\) Poisson NMF model fit.

fit_purified <-
  readRDS("../output/pbmc-purified/rds/fit-pbmc-purified-scd-ex-k=6.rds")$fit

Here, we explore the structure of the single-cell data as inferred by the topic model. Specifically, we use PCA to uncover structure in the estimated topic proportions of the multinomial topic model. Although PCA is simple, we will see that it works well, both for visualization and identifying clusters, and avoids the complications of the popular t-SNE and UMAP nonlinear dimensionality reduction methods. (Note that, since the topic proportions sum to 1, there are only 5 PCs to examine, not 6.)

fit <- poisson2multinom(fit_purified)
pca <- prcomp(fit$L)$x

Three large clusters are evident from first two PCs. We label the three large clusters as “A”, “B” and “C”. Since there are so many samples, the scatterplot suffers from “overplotting”, so it also helpful to view this PC projection as a density plot (“hex plot”).

n   <- nrow(pca)
x   <- rep("C",n)
pc1 <- pca[,"PC1"]
pc2 <- pca[,"PC2"]
x[pc1 + 0.2 > pc2] <- "A"
x[pc2 > 0.25] <- "B"
x[(pc1 + 0.4)^2 + (pc2 + 0.1)^2 < 0.07] <- "C"
samples_purified$cluster <- x
p1 <- pca_plot_with_labels(fit_purified,c("PC1","PC2"),
                           samples_purified$cluster) +
      labs(fill = "cluster")
p2 <- pca_hex_plot(fit_purified,c("PC1","PC2"))
plot_grid(p1,p2,rel_widths = c(9,10))

Version Author Date
7900d17 Peter Carbonetto 2020-08-22
38f07a2 Peter Carbonetto 2020-08-20

A small number of outlying data points do not seem to belong to any the three clusters, or they fall in between the clusters. For these data points, we assign them rather arbitrarily to one of the three clusters.

There also appears to be finer scale structure. We will examine this finer scale structure below.

pdat <- as.data.frame(pca)
p2 <- ggplot(pdat,aes(x = PC1,y = PC2,
             fill = cut(..count..,c(0,1,10,100,1000,Inf)))) +
  stat_bin_hex(bins = 40) +
  scale_fill_manual(values = c("gainsboro","lightskyblue","gold","orange",
                               "magenta")) +     
  labs(fill = "count") +
  theme_cowplot(font_size = 10)

Most of the samples are in cluster A:

table(x)
# x
#     A     B     C 
# 72614 10439 11602

Note that other PCs beyond the first two may also sometimes reveal additional clustering, and we will see examples of this in the 68k PBMC data.

Within cluster C there are two fairly well-defined subclusters (labeled “C1” and “C2”). There are perhaps other, less defined subclusters that are less defined, but in this analysis we focus on the largest, most obvious clusters.

rows <- which(samples_purified$cluster == "C")
fit  <- select(poisson2multinom(fit_purified),loadings = rows)
pca  <- prcomp(fit$L)$x
n    <- nrow(pca)
x    <- rep("C3",n)
pc1  <- pca[,1]
pc2  <- pca[,2]
x[pc1 < 0 & pc2 < 0.4] <- "C1"
x[pc1 > 0.5 & pc2 < 0.3] <- "C2"
samples_purified[rows,"cluster"] <- x
p2 <- pca_plot_with_labels(fit,c("PC1","PC2"),x) +
      labs(fill = "cluster")
print(p2)

Version Author Date
7900d17 Peter Carbonetto 2020-08-22

The two subclusters, C1 and C2, account for most of the samples in cluster C.

TO DO: Mention idea of a “background cluster”—a third subset of samples that do not fit well in the two clusters.

table(x)
# x
#   C1   C2   C3 
# 7822 2990  790

Now we turn to cluster A. Within this cluster, there is a large subcluster, which we label as “A1”. (This cluster is much less distinct than the other clusters we have seen so far, and may not show up clearly in this plot—you may need to zoom in on the plot to see the clustering.) Otherwise, there is no obvious additional clustering of the samples within cluster A.

rows <- which(samples_purified$cluster == "A")
fit  <- select(poisson2multinom(fit_purified),loadings = rows)
pca  <- prcomp(fit$L)$x
n    <- nrow(fit$L)
x    <- rep("A2",n)
pc1  <- pca[,1]
pc2  <- pca[,2]
x[pc1 > 0.58 - pc2 | pc1 > 0.7] <- "A1"
samples_purified[rows,"cluster"] <- x
p3 <- pca_plot_with_labels(fit,c("PC1","PC2"),x) +
      labs(fill = "cluster")
print(p3)

Version Author Date
97d7e86 Peter Carbonetto 2020-08-23
7900d17 Peter Carbonetto 2020-08-22

In summary, we have subdivided the data into 6 subsets:

samples_purified$cluster <- factor(samples_purified$cluster)
table(samples_purified$cluster)
# 
#    A1    A2     B    C1    C2    C3 
#  8271 64343 10439  7822  2990   790

We also inspected principal components individually in each of these 6 clusters and we did not find any of clear examples of subclustering withing these clusters.

The structure plot summarizes the topic proportions in each of these 6 subsets:

set.seed(1)
pbmc_purified_topic_colors <- c("gold","forestgreen","dodgerblue",
                                "gray","greenyellow","magenta")
pbmc_purified_topics <- c(2,5,3,1,4,6)
rows <- sort(c(sample(which(samples_purified$cluster == "A1"),250),
               sample(which(samples_purified$cluster == "A2"),1200),
               sample(which(samples_purified$cluster == "B"),250),
               sample(which(samples_purified$cluster == "C1"),250),
               sample(which(samples_purified$cluster == "C2"),200),
               sample(which(samples_purified$cluster == "C3"),200)))
p4 <- structure_plot(select(poisson2multinom(fit_purified),loadings = rows),
                     grouping = samples_purified[rows,"cluster"],
                     topics = pbmc_purified_topics,
                     colors = pbmc_purified_topic_colors[pbmc_purified_topics],
                     n = Inf,perplexity = c(70,100,70,70,50,50),
                     gap = 40,num_threads = 4,verbose = FALSE)
print(p4)

Version Author Date
eac2d23 Peter Carbonetto 2020-08-25
2d156b8 Peter Carbonetto 2020-08-25
abb846e Peter Carbonetto 2020-08-25
f53c86c Peter Carbonetto 2020-08-24
13ee038 Peter Carbonetto 2020-08-23
97d7e86 Peter Carbonetto 2020-08-23
59777e7 Peter Carbonetto 2020-08-22
c87ddf8 Peter Carbonetto 2020-08-22
7900d17 Peter Carbonetto 2020-08-22
fbb0697 Peter Carbonetto 2020-08-21
216027a Peter Carbonetto 2020-08-21

Out of the 6 topics, 4 of them (\(k = 2, 3, 4, 5\)) align closely with clusters (labeled A1, B, C1, C2). And, indeed, they align closely with their inclusion in the individual FACS-purified data sets:

pdat <- as.data.frame(with(samples_purified,table(celltype,cluster)))
p5 <- ggplot(pdat,aes(x = cluster,y = celltype,size = Freq)) +
  geom_point(color = "dodgerblue",na.rm = TRUE,show.legend = FALSE) +
  ylim(rev(levels(samples_purified$celltype))) +
  theme_cowplot(font_size = 10)
print(p5)

Version Author Date
eac2d23 Peter Carbonetto 2020-08-25
2d156b8 Peter Carbonetto 2020-08-25
abb846e Peter Carbonetto 2020-08-25

Based on the above results, we make a few observations:

In summary, a cluster-based analysis and topic-based analysis should yield mostly similar results, except for the analysis of cluster A2, which should benefit from a topic-based analysis (specifically, analysis of topics 1 and 6).

Unsorted 68k PBMC data

Next, we turn to the 68k data set.

load("../data/pbmc_68k.RData")
samples_68k <- samples
rm(samples,genes,counts)

Load the \(k = 6\) Poisson NMF model fit, and compute PCs from the topic proportions.

fit_68k <- readRDS("../output/pbmc-68k/rds/fit-pbmc-68k-scd-ex-k=6.rds")$fit
fit <- poisson2multinom(fit_68k)
pca <- prcomp(fit$L)$x

In this case, we find least three distinct clusters in the projection onto PCs 3 and 4. We label these clusters “A”, “B” and “C”, as above, noting that this labeling does not imply a connection with the purified PBMC clusters above.

n <- nrow(pca)
x <- rep("A",n)
pc3 <- pca[,"PC3"]
pc4 <- pca[,"PC4"]
x[pc4 < -0.13 | pc3/1.9 - 0.17 > pc4] <- "B"
x[pc4 < -0.75] <- "C"
samples_68k$cluster <- x
p6 <- pca_plot_with_labels(fit_68k,c("PC3","PC4"),x) +
      labs(fill = "cluster")
print(p6)

Version Author Date
f53c86c Peter Carbonetto 2020-08-24
a406a2f Peter Carbonetto 2020-08-22
7900d17 Peter Carbonetto 2020-08-22
fbb0697 Peter Carbonetto 2020-08-21
216027a Peter Carbonetto 2020-08-21
6d3d7e4 Peter Carbonetto 2020-08-20

The vast majority of the cells are in cluster A.

table(samples_68k$cluster)
# 
#     A     B     C 
# 63408  5006   165

Looking more closely at the top two PCs in cluster B, we identify two large clusters, with the remaining samples assigned to the “B3” subset.

rows <- which(samples_68k$cluster == "B")
fit  <- select(poisson2multinom(fit_68k),loadings = rows)
pca  <- prcomp(fit$L)$x
n    <- nrow(pca)
x    <- rep("B3",n)
pc1  <- pca[,"PC1"]
x[pc1 > -0.05] <- "B1"
x[pc1 < -0.3] <- "B2"
samples_68k[rows,"cluster"] <- x
p7 <- pca_plot_with_labels(fit,c("PC1","PC2"),x) +
      labs(fill = "cluster")
print(p7)

Version Author Date
f53c86c Peter Carbonetto 2020-08-24
b6489db Peter Carbonetto 2020-08-23

The B1 cluster can be further subdivided into clusters (B1a, B1b), with subset B1c compresing the B1 samples that do not fit into either cluster.

rows <- which(samples_68k$cluster == "B1")
fit  <- select(poisson2multinom(fit_68k),loadings = rows)
pca  <- prcomp(fit$L)$x
n    <- nrow(pca)
x    <- rep("B1c",n)
pc1  <- pca[,"PC1"]
pc2  <- pca[,"PC2"]
x[pc2 > -0.02 & pc1 > -0.25] <- "B1a"
x[pc1 > 0.1 & pc2 < 0.05 & pc2 > -0.225] <- "B1b"
samples_68k[rows,"cluster"] <- x
p8 <- pca_plot_with_labels(fit,c("PC1","PC2"),x) +
      labs(fill = "cluster")
print(p8)

Version Author Date
399c597 Peter Carbonetto 2020-08-25
f53c86c Peter Carbonetto 2020-08-24
b6489db Peter Carbonetto 2020-08-23

Cluster A subdivides fairly neatly into two large clusters, A1 and A2.

rows <- which(samples_68k$cluster == "A")
fit  <- select(poisson2multinom(fit_68k),loadings = rows)
pca  <- prcomp(fit$L)$x
n    <- nrow(pca)
x    <- rep("A3",n)
pc2  <- pca[,"PC2"]
pc3  <- pca[,"PC3"]
x[2.5*pc3 < 0.3 - pc2] <- "A1"
x[pc3 > 0.8 - pc2] <- "A2"
samples_68k[rows,"cluster"] <- x
p9 <- pca_plot_with_labels(fit,c("PC2","PC3"),x) +
      labs(fill = "cluster")
print(p9)

Version Author Date
399c597 Peter Carbonetto 2020-08-25

Within cluster A, the vast majority of the samples are assigned to the A1 subcluster:

table(x)
# x
#    A1    A2    A3 
# 59260  3555   593

In summary, we have subdivided these data into 9 subsets:

samples_68k$cluster <- factor(samples_68k$cluster)
table(samples_68k$cluster)
# 
#    A1    A2    A3   B1a   B1b   B1c    B2    B3     C 
# 59260  3555   593  2115   947   807   819   318   165

The wide range in the sizes of these clusters is notable; the smallest cluster (C) is less than 1% the size of the largest (A1). By contrast, community detection methods such as the Louvain algorithm are preferentially biased toward more uniformly sized clusters (this is a known limitation of community detection methods).

The structure plot summarizes the topic proportions in each of these 9 subsets:

set.seed(1)
pbmc_68k_topic_colors <- c("yellow","lightskyblue","salmon",
                           "firebrick","royalblue","olivedrab")
pbmc_68k_topics <- c(2,5,1,3,4,6)
rows <- sort(c(sample(which(samples_68k$cluster == "A1"),1200),
               sample(which(samples_68k$cluster == "A2"),500),
               sample(which(samples_68k$cluster == "A3"),300),
               sample(which(samples_68k$cluster == "B1a"),500),
               sample(which(samples_68k$cluster == "B1b"),300),
               sample(which(samples_68k$cluster == "B1c"),300),
               sample(which(samples_68k$cluster == "B2"),300),
               which(samples_68k$cluster == "B3"),
               which(samples_68k$cluster == "C")))
p10 <- structure_plot(select(poisson2multinom(fit_68k),loadings = rows),
                      grouping = samples_68k[rows,"cluster"],
                      topics = pbmc_68k_topics,
                      colors = pbmc_68k_topic_colors[pbmc_68k_topics],
                      perplexity = c(100,100,50,100,80,80,80,80,50),
                      n = Inf,gap = 32,num_threads = 4,verbose = FALSE)
print(p10)

Version Author Date
399c597 Peter Carbonetto 2020-08-25
abb846e Peter Carbonetto 2020-08-25
f53c86c Peter Carbonetto 2020-08-24
a0cb7c6 Peter Carbonetto 2020-08-24
7900d17 Peter Carbonetto 2020-08-22
fbb0697 Peter Carbonetto 2020-08-21
216027a Peter Carbonetto 2020-08-21

These subsets do not align as closely with the cell-type labeling inferred by Zheng et al (2017), which is not surprising considering that this labeling is based on the FACS-purified data set.

pdat <- as.data.frame(with(samples_68k,table(celltype,cluster)))
p11 <- ggplot(pdat,aes(x = cluster,y = celltype,size = Freq)) +
  geom_point(color = "dodgerblue",na.rm = TRUE,show.legend = FALSE) +
  ylim(rev(levels(samples_purified$celltype))) +
  theme_cowplot(font_size = 10)
print(p11)

A few notes about these results:

In summary, the topics and clusters seem to offer very much complementary biological insights, although subsequent analysis is needed to determine what these insights are.


sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.5
# 
# 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] cowplot_1.0.0      ggplot2_3.3.0      fastTopics_0.3-165 dplyr_0.8.3       
# 
# loaded via a namespace (and not attached):
#  [1] ggrepel_0.9.0        Rcpp_1.0.5           lattice_0.20-38     
#  [4] tidyr_1.0.0          prettyunits_1.1.1    assertthat_0.2.1    
#  [7] zeallot_0.1.0        rprojroot_1.3-2      digest_0.6.23       
# [10] R6_2.4.1             backports_1.1.5      MatrixModels_0.4-1  
# [13] evaluate_0.14        coda_0.19-3          httr_1.4.1          
# [16] pillar_1.4.3         rlang_0.4.5          progress_1.2.2      
# [19] lazyeval_0.2.2       data.table_1.12.8    irlba_2.3.3         
# [22] SparseM_1.78         hexbin_1.28.0        whisker_0.4         
# [25] Matrix_1.2-18        rmarkdown_2.3        labeling_0.3        
# [28] Rtsne_0.15           stringr_1.4.0        htmlwidgets_1.5.1   
# [31] munsell_0.5.0        compiler_3.6.2       httpuv_1.5.2        
# [34] xfun_0.11            pkgconfig_2.0.3      mcmc_0.9-6          
# [37] htmltools_0.4.0      tidyselect_0.2.5     tibble_2.1.3        
# [40] workflowr_1.6.2.9000 quadprog_1.5-8       viridisLite_0.3.0   
# [43] crayon_1.3.4         withr_2.1.2          later_1.0.0         
# [46] MASS_7.3-51.4        grid_3.6.2           jsonlite_1.6        
# [49] gtable_0.3.0         lifecycle_0.1.0      git2r_0.26.1        
# [52] magrittr_1.5         scales_1.1.0         RcppParallel_5.0.2  
# [55] stringi_1.4.3        farver_2.0.1         fs_1.3.1            
# [58] promises_1.1.0       vctrs_0.2.1          tools_3.6.2         
# [61] glue_1.3.1           purrr_0.3.3          hms_0.5.2           
# [64] yaml_2.2.0           colorspace_1.4-1     plotly_4.9.2        
# [67] knitr_1.26           quantreg_5.54        MCMCpack_1.4-5