# Cop1 role in pro-inflammatory response

ICME, Stanford University, CA 94305

# Abstract

The analysis of time-series data is non-trivial as in involves dependencies between datapoints. Our package helps researchers analyze RNA-seq datasets which include gene expression measurements taken over time. The methods are specifically designed for at datasets with a small number of replicates and time points – a typical case for RNA-seq time course studies. Short time courses are more difficult to analyze, as many statistical methods designed for time-series data might require a minimum number of time points, e.g. functional data analysis (FDA) and goodness of fit methods might be ineffective. Our approach is non-parametric and gives the user a flexibility to incorporate different normalization techniques, and distance metrics. TimeSeriesExperiment is a comprehensive time course analysis toolbox with methods for data visualization, clustering and differential expression analysis. Additionally, the package can perform enrichment analysis for DE genes.

# Introduction

We will demonstrate the effectiveness of TimeSeriesExperiment package using the in-house dataset exploring the role of Cop1 in pro-inflammatory response. The experiments were designed to study the induction and repression kinetics of genes in bone marrow derived macrophages (BMDMs).

The dataset includes cells from 6 mice. The cells were divided into two equal groups. For one group the Cop1 gene was in vitro knock out with tamoxifen. All samples where then subject to LPS treatment to induce an inflammatory response. Bulk RNA-seq was performed at 6 time-points: one at time 0 before LPS treatment, then at time 2.5, 4, 6, 9 and 13 hours after LPS was added.

# Obtain and process data

First we load the necessary packages.

##                edgeR              viridis              Biobase SummarizedExperiment              ggplot2                dplyr                tidyr               tibble                readr
##                 TRUE                 TRUE                 TRUE                 TRUE                 TRUE                 TRUE                 TRUE                 TRUE                 TRUE
## TimeSeriesExperiment
##                 TRUE

The dataset we will study is available from GEO repositories under accession number: GSE114762. We can import the processed read counts saved in a supplementary file ‘GSE114762_raw_counts.csv.gz’. Similarly both the phenotype and feature data are also available for download as ‘GSE114762_sample_info.csv.gz’, and ‘GSE114762_gene_data.csv.gz’ files respectively.

# Building TimeSeriesExperiment object

We can now combine all the data, cnts, gene.data and pheno.data, into a single TimeSeriesExperiment (S4) object. The data will store everything together in a way that is easier to perform further time course data analysis. The most important fields in the object are “assays”, “colData” which will contain information on group, replicate and time associated with each sample. In TimeSeriesExperiment the time variable MUST be numeric.

## class: TimeSeriesExperiment
## dim: 36528 36
## assays(1): raw
## rownames(36528): 100009600 100009609 ... 99929 99982
## rowData names(5): feature symbol size type desc
## colnames(36): SAM24331086 SAM24331087 ... SAM24331084 SAM24331085
## colData names(9): sample group ... timepoint category
## ==========
## timepoints(36): 0 2.5 ... 9 13
## groups(36): Loxp Loxp ... WT WT
## replicates(36): Loxp_1 Loxp_1 ... WT_3 WT_3
## -----
## assayCollapsed: NULL
## colDataCollapsed: NULL
## timeSeries: NULL
## dimensionReduction: NULL
## clusterRows: NULL
## differentialExpression: NULL

Alternatively, we can build TimeSeriesExperiment object from a ExpressionSet or SummarizedExperiment objects.

To show how this is done, we first combine the three data tables into an ExpressionSet objects.

Then, we can easily convert an ExpressionSet to a TimeSeriesExperiment object using makeTimeSeriesExperimentFromExpressionSet() function, and indicating columns names with relevant information.

## class: TimeSeriesExperiment
## dim: 36528 36
## assays(1): raw
## rownames(36528): 100009600 100009609 ... 99929 99982
## rowData names(5): feature symbol size type desc
## colnames(36): SAM24331086 SAM24331087 ... SAM24331084 SAM24331085
## colData names(9): sample group ... timepoint category
## ==========
## timepoints(36): 0 2.5 ... 9 13
## groups(36): Loxp Loxp ... WT WT
## replicates(36): 1 1 ... 3 3
## -----
## assayCollapsed: NULL
## colDataCollapsed: NULL
## timeSeries: NULL
## dimensionReduction: NULL
## clusterRows: NULL
## differentialExpression: NULL

Repeating the same for SummarizedExperiment:

## class: TimeSeriesExperiment
## dim: 36528 36
## assays(1): raw
## rownames(36528): 100009600 100009609 ... 99929 99982
## rowData names(5): feature symbol size type desc
## colnames(36): SAM24331086 SAM24331087 ... SAM24331084 SAM24331085
## colData names(9): sample group ... timepoint category
## ==========
## timepoints(36): 0 2.5 ... 9 13
## groups(36): Loxp Loxp ... WT WT
## replicates(36): 1 1 ... 3 3
## -----
## assayCollapsed: NULL
## colDataCollapsed: NULL
## timeSeries: NULL
## dimensionReduction: NULL
## clusterRows: NULL
## differentialExpression: NULL

## Data normalization and filtering

The raw read counts cannot be immediately used for analysis, as sequencing data involves the issue of varying sample depths. We can convert the raw counts to counts per million (CPM) using TimeSeriesExperiment function, normalizeData(), which performs data normalization by column. Currently, we only support scaling sample counts by constant factors (size factors). If the argument column.scale.factor is not specified, by default TimeSeriesExperiment divides by column sums and multiplies by 1 million to obtain CPMs. The normalized data is stored separately from the raw data in a slot “data”.

## Normalizing data...

Since the dataset contains more than 36k genes, we will filter out the very rare ones which we assume to be too noisy and not containing enough signal for further analysis.

Here, we find and remove all genes which have the mean expression (CPM) below min_mean_cpm = 50 within either of the two groups of interest, wild type or knock-out. We set a very large threshold of 100 for this vignette only because the graphics below would make the size of this vignette too large. When running on the your own study you should pick the threshold more carefully.

## class: TimeSeriesExperiment
## dim: 9356 36
## assays(2): raw norm
## rownames(9356): 100017 100019 ... 99929 99982
## rowData names(5): feature symbol size type desc
## colnames(36): SAM24331086 SAM24331087 ... SAM24331084 SAM24331085
## colData names(9): sample group ... timepoint category
## ==========
## timepoints(36): 0 2.5 ... 9 13
## groups(36): Loxp Loxp ... WT WT
## replicates(36): 1 1 ... 3 3
## -----
## assayCollapsed: NULL
## colDataCollapsed: NULL
## timeSeries: NULL
## dimensionReduction: NULL
## clusterRows: NULL
## differentialExpression: NULL

## Collapse replicates

In parts of our later analysis, we will make comparisons between genes, and therefore it is useful to aggregate gene expression across replicates to obtain their mean behavior. To do this we can use collapseReplicates() function for TimeSeriesExperiment. The function saves collapsed sample and aggregated expression data in “sample.data.collapsed”, and “data.collapsed” respectively.

## Aggregating across replicates...

# Time course format

The main focus on TimeSeriesExperiment is to analyze and visualize time-series data efficiently. For this reason, we convert the expression data in a form of a rectangular matrix into a “time-course format” where each row stores a single time series corresponding to a specified combination of group membership and and replicate id (here mouse id). This data wrangling step can be performed with makeTimeSeries() function, the “time-course” will be stored in a slot timeSeries. This slot contains a list containing data.frames tc and tc_collapsed (if assayCollapsed is defined).

Before converting data to “time-course” format, gene transformation should be performed. Transformation is thus allows for a fair gene-to-gene comparison. For example, when clustering genes, one uses pairwise distances to estimate dissimilarities between genes. Since, we are generally more interested in grouping genes based on similarities in their trajectories rather than their absolute expression levels, the read counts must be transformed before computing the distances.

Currently, gene transformation methods available in TimeSeriesExperiment are “scale_feat_sum” (scaling by gene sum) or “var_stab” (variance stabilization). The user can specify a variance stabilization method if “var_stab” is used. VST methods supported are: “log1p” (log plus one), “asinh” (inverse hyperbolic sine) or “deseq” (DESeq2::varianceStabilizingTransformation).

Usually simply scaling by the gene sum, that is normalizing so that the total abundance the same (and equal to 1) for all genes gives good clustering of gene trajectories.

## Converting to timeseries format...
feature group replicate 0 2.5 4 6 9 13
100017 Loxp 1 181.2367 30.48920 13.41001 19.58618 48.43795 157.1302
100017 Loxp 2 188.6391 30.57038 14.31970 29.69035 83.93528 168.5095
100017 Loxp 3 193.7659 27.61548 12.98989 23.28758 51.96757 155.0601
100017 WT 1 186.9893 27.54414 13.56621 22.96282 52.04936 148.6426
100017 WT 2 220.9431 32.70219 11.50873 20.85574 45.29843 161.3555
100017 WT 3 200.6372 23.68253 12.28389 23.77404 51.05345 152.3733
feature group replicate 0 2.5 4 6 9 13
100017 Loxp 1 0.0638413 0.0107399 0.0047237 0.0068993 0.0170624 0.0553497
100017 Loxp 2 0.0664488 0.0107685 0.0050442 0.0104585 0.0295665 0.0593581
100017 Loxp 3 0.0682547 0.0097277 0.0045757 0.0082031 0.0183058 0.0546205
100017 WT 1 0.0658677 0.0097025 0.0047787 0.0080887 0.0183346 0.0523599
100017 WT 2 0.0778280 0.0115195 0.0040540 0.0073465 0.0159565 0.0568380
100017 WT 3 0.0706752 0.0083423 0.0043270 0.0083745 0.0179838 0.0536741

## Lag differences

Time-series data have a dependency structure and are not standard multivariate vectors. Many methods have been developed for representing time-series data. A common technique is for example to fit functions e.g. polynomials or splines to the data. A similar approach is taken in functional data analysis (FDA) literature, where time series are represented as linear combinations of basis functions (e.g. principal functions). These methods seek to smooth the data and to reduce of complexity of the inherent (infinite dimensional) functional data. The fitted coefficients are often used as the time-series representation then used for clustering or visualization.

Unfortunately, most of the biological time course studies are short, sometimes containing as few as three or four time-points. Therefore, fitting functions to sparse time points would be inefficient. Instead, here we propose a simpler way to incorporate the dependency structure of the time-series. Our method involves construction of additional data features, which are lag differences between consecutive time-points. Lag of order 1 for time-point $$i$$, $$Lag_1(i)$$, denotes the difference between the expression level at time $$i$$ and time $$i-1$$, lag 2 is the difference between time $$i$$ and time $$i-2$$, and so on. Intuitively, the lag 1 at time $$i$$ approximates the slope or the first derivative of the time series curve at time-point $$i$$.

We can add these extra lag features to the “time-course” data using addLags() function, which appends lag features to “tc” and “tc_collapsed” data frames in the slot timeSeries. Additionally,the user can define the weight for each lag feature by specifying the lambda argument. The length of lambda indicates how many orders of lags you would like to include, e.g. lambda = c(0.5, 0.25) means lag order 1 will be added with multiplicative weight of $$0.5$$ and lag order 2 will be added with weight $$0.025$$.

## Adding lags with coefficients: 0.5 0.25...
feature group replicate 0 2.5 4 6 9 13 Lag_2.5_0 Lag_4_2.5 Lag_6_4 Lag_9_6 Lag_13_9 Lag_4_0 Lag_6_2.5 Lag_9_4 Lag_13_6
100017 Loxp 1 0.0638413 0.0107399 0.0047237 0.0068993 0.0170624 0.0553497 -0.0265507 -0.0030081 0.0010878 0.0050816 0.0191436 -0.0147794 -0.0009602 0.0030847 0.0121126
100017 Loxp 2 0.0664488 0.0107685 0.0050442 0.0104585 0.0295665 0.0593581 -0.0278401 -0.0028622 0.0027072 0.0095540 0.0148958 -0.0153512 -0.0000775 0.0061306 0.0122249
100017 Loxp 3 0.0682547 0.0097277 0.0045757 0.0082031 0.0183058 0.0546205 -0.0292635 -0.0025760 0.0018137 0.0050513 0.0181574 -0.0159198 -0.0003811 0.0034325 0.0116043
100017 WT 1 0.0658677 0.0097025 0.0047787 0.0080887 0.0183346 0.0523599 -0.0280826 -0.0024619 0.0016550 0.0051229 0.0170127 -0.0152722 -0.0004034 0.0033890 0.0110678
100017 WT 2 0.0778280 0.0115195 0.0040540 0.0073465 0.0159565 0.0568380 -0.0331543 -0.0037327 0.0016463 0.0043050 0.0204408 -0.0184435 -0.0010432 0.0029756 0.0123729
100017 WT 3 0.0706752 0.0083423 0.0043270 0.0083745 0.0179838 0.0536741 -0.0311665 -0.0020076 0.0020237 0.0048046 0.0178452 -0.0165870 0.0000081 0.0034142 0.0113249

At this point, we completed all data pre-processing steps available in TimeSeriesExperiment. In later sections we specify how visualization, clustering and differential expression test can be performed with the package.

# Data Visualization

In this section we show plotting utilities available in TimeSeriesExperiment. Visualizations are data exploration tools and serve as the first step in our data analysis. In the following subsections we will describe more in details how heatmaps and PCA plots can be generated.

## Heatmaps

Here we will generate a heatmap of top 100 most variable genes. The plot of the expression matrix for these most variable features will give us some insight whether there is a clear difference between the two experimental groups and whether a strong temporal trend can be detected.

In the above heatmap the columns are ordered by experimental group, replicate (mouse id) and time at which the sample was sequenced; the sample membership is indicated in the color bars on top of the columns. The main heatmap rectangle shows Z-scores of expression values represented by colors in the red-and-blue palette corresponding to high-and-low respectively. Even this first look at the data, shows us patterns present in the data – within each condition, i.e. each mouse and in each experimental group there are expression levels seem to be dependent on time.

# PCA

Another way to explore the dataset is through dimensionality reduction. Here we will project the data into a space of principal components. With PCA, you can visualize both samples and features in the same coordinates space with a biplot. Here we will keep these two maps separately, as the visualization can become overcrowded with points which obscure the inherent structure. Even though, we are plotting the feature and sample projections separately, they can be compared side by side to see which groups of features are more correlated with which group of samples.

## Visualizing Samples

First, we project samples on a 2D map to check whether their relative location reflects time at which the sequencing was performed. If the samples are ordered in agreement with time in the PCA plot, there might be patterns in gene expression levels changing over the course of the study.

A PCA plot can also be used to examine whether samples corresponding to the same condition, here wild type or knockouts, tend to group together, i.e. whether they are more similar to each other than to the ones in a different condition. We use prcomp() function from stats (default) package to compute PCA projection.

RNA-seq data is highly heteroscedastic, which means the features (genes) included have vastly different variances. It is know that bulk expression count data can be well modeled with a negative binomial distribution. In this distribution, variance is a quadratic function of the mean, $$\sigma^2 = \mu + \alpha\mu^2$$. That is the higher the mean expression level, the higher the variance is. PCA projection maximizes the amount variance preserved in consecutive principal components, which implies that computing PCA on raw expression counts or even the RPKMs or CPMs would put too much weight on most highly abundant genes.

This step is recommended because RNA-seq data is highly heteroscedastic, which means the features (genes) included have vastly different variances. It is know that bulk expression count data can be well modeled with a negative binomial distribution. In this distribution, variance is a quadratic function of the mean, $$\sigma^2 = \mu + \alpha\mu^2$$. That is the higher the mean expression level, the higher the variance is.

Thus, we recommend user to variance stabilize the data before computing a PCA projection (as described in Time course format section). The variance stabilization method can be specified in var.stabilize.method argument.

Additionally, the user might limit calculations to only specific group of samples, e.g. in this case you might be interested in visualizing samples only in the wild type group. A user can also indicate whether PCA should be applied to sample resolved data or one with replicates aggregated (stored in “data.collapsed” slot of a TimeSeriesExperiment object).

In the plots above we see that the samples group mostly by time at which they have been sequenced. For some time-points, we see also a clear separation between sample from different experimental groups.

## Visualizing Features

PCA also provides a projection of features to the same principal component space. The coordinates of features (here genes) are commonly referred to as PCA loadings. Since gene expression datasets usually includes thousands of genes, it is not possible to include labels for all of them in the same 2D PCA plot. Apart from that, in a time-course study one is usually interested in trajectories of genes over time, and it is good to see groups of genes with similar expression pattern clustered together on a visualization.

In order to make PCA plots for features more informative, we overlay average (over replicates) gene expression trend over time for each experimental group. Plotting trajectories for every gene would make plot overcrowded and unreadable, therefore we divide the PCA plot into $$m \times n$$ grid and plot a trajectory for a gene which PCA coordinates are closes to the grid center point.

In the feature PCA plot below we see that genes exhibit different responses to LPS treatment. The figure shows genes organized according to their trajectory. Inhibited genes tend to gather on the left side of the plot, the primary response (early spike) genes are at the bottom, and the late response (late increase) clustered at the top. The plot doesn’t show a global difference between the wild type and knock-out group. Most genes have trajectories that are exactly overlapping in both conditions. There are, however, a few genes for which we do observe some difference between groups.

# Gene Clustering

To cluster the gene trajectories we will use the data stored in timeSeries slot of cop1.te. These are time-series of transformed expression values together with appended time lags which resolving differences between the temporal trends.

We use hierarchical clustering define gene groupings. Either static or dynamic branch cutting (from dynamicTreeCut package) algorithms can be used to assign clusters. Since hierarchical clustering is computationally intensive (with $$O(n^3)$$ complexity for standard implementations), we apply it only to a subset of genes. Specifically, we pick n.top.feat with average (over replicates) most variable expression over time in each of selected groups (here we use both wild type and knock-out) to perform clustering. Remaining genes are, then, assigned to a cluster with the closest centroid. An additional advantage of using only a subset of most variable genes for clustering is that, the core genes which exhibit negligible changes over time (which might be the majority of genes) will not much effect on clustering results.

Here we pick 1000 genes for clustering, which is roughly 1/3 of the number of genes after filtering.

## Averaging timecourses over all 'groups' selected and recomputing lags with coefficients: 0.5 0.25

We can see the size of each of clusters computed

##
##   C1   C2   C3   C4   C5   C6
## 2873 2418 2201  979  478  407

We can plot the hclust dendrogram obtained from hierarchical clustering performed.

Here we plot average (over replicates) gene trajectories grouped into 10 clusters found using the above described approach. The expression profiles for wild type and knock-out are plotted separately, side by side.

Timecourse plots above show genes clustered clearly into groups related to their pattern of response to LPS treatment. We see cluster C1, C2, and C4 generally inhibited. Genes in C6 spike right after LPS was applied. C3 shows moderate secondary response, and C5 exhibit late increase in expression.

# Differential Expression Ranking

In the previous section we grouped the genes based on the trajectories recorded for both experimental groups. In this section we will describe how to find specific genes that exhibit different expression patterns between two experimental groups over the time-course of the study.

## Differential Point-wise Expression

In some cases, a user might be interested in differential expression (DE) at specific timepoints between different experimental groups, here wild type and knock-out. We can easily test differential expression at any timepoint over the course of the study using standard DE approaches.

TimeSeriesExperiment provides a wrapper timepointDE() for differential expression testing functions (voom() + limma()) from limma package, which allows users to easily apply testing to TimeSeriesExperiment objects.

## testing timepoint: 0
## testing timepoint: 2.5
## testing timepoint: 4
## testing timepoint: 6
## testing timepoint: 9
## testing timepoint: 13

Information on DE genes e.g. at timepoint 2.5 can be access as follows:

feature symbol size type desc cluster used_for_hclust logFC AveExpr t P.Value adj.P.Val B
12816 12816 Col12a1 14801 protein_coding collagen, type XII, alpha 1 C1 TRUE 1.6133178 3.766775 12.408537 6.0e-07 0.0026451 6.533085
26374 26374 Rfwd2 5341 protein_coding ring finger and WD repeat domain 2 C3 FALSE 1.2880720 3.273972 12.518509 6.0e-07 0.0026451 6.473224
213002 213002 Ifitm6 541 protein_coding interferon induced transmembrane protein 6 C6 TRUE -1.7168075 2.308143 -11.984034 8.0e-07 0.0026451 5.673224
320832 320832 Sirpb1a 1786 protein_coding signal-regulatory protein beta 1A C6 TRUE -0.9648697 3.456528 -10.529209 2.5e-06 0.0058671 5.279358
109648 109648 Npy 547 protein_coding neuropeptide Y C2 FALSE -0.8567081 6.071158 -10.098207 3.5e-06 0.0066364 5.060502
13039 13039 Ctsl 2327 protein_coding cathepsin L C3 FALSE -0.6281533 11.877995 -9.649871 5.2e-06 0.0066451 4.676455

To find genes with the highest log-fold change in each timepoint we can call the following commands:

0 2.5 4 6 9 13
NA Itga11 Col2a1 Col2a1 Wisp1 Thbs2
NA Postn Mpo Actg2 Timp3 Postn
NA Col1a1 Itga11 Gm22 Postn Col1a1
NA Actg2 Gm22 Col1a1 Thbs2 Timp3
NA Timp3 Crlf1 Lrrc32 Col1a1 Col1a2
NA Wisp1 Col1a1 Wisp1 Hspg2 Rcn3
NA Cxcl5 Wisp1 Timp3 Vgll3 Col3a1
NA Crlf1 Timp3 Itga11 Crlf1 Hspg2
NA Ddr1 Hspg2 Hspg2 Myl9 Crlf1
NA Scube3 Thbs2 Ddr1 Grem2 Col5a1

We can also find a list of genes which were found differentially expressed at any of the timepoints

## Out of all 9356 , there were 997 were found differentially expressed at any timepoint.

A useful diagram would show intersections of differentially expressed genes at different time-points. You should expert more intersection between consecutive time-points. You can use functions from UpSetR package to show the overlap between the DE genes across timepoints:

## Differential Trajectories

Instead of looking at each time-point separately, it is often useful to identifying genes which exhibit different expression trajectories, i.e. ones with differential kinetics over time. We take a most natural approach which can be applied to short time-course datasets, which is an analysis of variance. In particular we use a method based on non-parametric permutation multivariate analysis of variance (MANOVA).

To test each gene for differential trajectory under two conditions, we use the time-course format, where each row is a transformed time-series with lags corresponding to a single replicate within a particular group. adonis() function from vegan package is applied to find genes with differential trajectories. adonis approach is based on partitioning the sums of squares of the multivariate (here time-series with lags) data to within and between-class. The significance is determined with an F-test on permutations of the data.

Using this procedure, here we will identify genes which have different trajectories within the wild type and knock-out group. This difference is determined when time series replicates (expression profiles) are more different between the groups than within the same group.

With a small number of available replicates (3 wild type and 3 knockout), a permutation based method does not yield high power. Combined with multiple hypothesis testing correction (for testing thousands of genes), we expect the method’s p-values to be mostly below significance level of $$\alpha = 0.5$$. However, this approach is still useful, as we can user raw (unadjusted) p-values to filter out the genes that are with high probability not significant, and use the $$R^2$$ value (the percentage variance explained by groups) for ranking the genes in terms of the difference in expression profiles between two groups.

Function trajectoryDE() can be used to find differential genes using the above described method. Results of testing procedure can be accessed with: differentialExpression(cop1.te, "trajectory_de").

feature Df SumsOfSqs MeanSqs F.Model R2 pval p.adj symbol size type desc cluster used_for_hclust
26374 1 0.0034744 0.0034744 112.51257 0.9656689 0.1 0.3534567 Rfwd2 5341 protein_coding ring finger and WD repeat domain 2 C4 FALSE
320782 1 0.0072386 0.0072386 103.81930 0.9629009 0.1 0.3534567 Tmem154 3342 protein_coding transmembrane protein 154 C2 TRUE
104156 1 0.0048415 0.0048415 83.55940 0.9543167 0.1 0.3534567 Etv5 4066 protein_coding ets variant 5 C6 TRUE
27355 1 0.0118039 0.0118039 65.46214 0.9424147 0.1 0.3534567 Pald1 4490 protein_coding phosphatase domain containing, paladin 1 C2 TRUE
16763 1 0.0087209 0.0087209 63.29198 0.9405576 0.1 0.3534567 Lad1 2992 protein_coding ladinin C3 TRUE
13039 1 0.0011924 0.0011924 57.83038 0.9353069 0.1 0.3534567 Ctsl 2327 protein_coding cathepsin L C4 FALSE
17394 1 0.0039649 0.0039649 56.47761 0.9338598 0.1 0.3534567 Mmp8 2453 protein_coding matrix metallopeptidase 8 C4 TRUE
66455 1 0.0015594 0.0015594 43.69125 0.9161272 0.1 0.3534567 Cnpy4 1781 protein_coding canopy 4 homolog (zebrafish) C1 TRUE
240913 1 0.0040984 0.0040984 43.51711 0.9158198 0.1 0.3534567 Adamts4 3673 protein_coding a disintegrin-like and metallopeptidase (reprolysin type) with thrombospondin type 1 motif, 4 C6 TRUE
219144 1 0.0011053 0.0011053 41.02111 0.9111528 0.1 0.3534567 Arl11 2810 protein_coding ADP-ribosylation factor-like 11 C2 TRUE
15370 1 0.0111295 0.0111295 40.50848 0.9101295 0.1 0.3534567 Nr4a1 5035 protein_coding nuclear receptor subfamily 4, group A, member 1 C2 TRUE
213956 1 0.0074680 0.0074680 38.03157 0.9048334 0.1 0.3534567 Fam83f 3317 protein_coding family with sequence similarity 83, member F C2 TRUE
15530 1 0.0127704 0.0127704 34.42740 0.8959076 0.1 0.3534567 Hspg2 15128 protein_coding perlecan (heparan sulfate proteoglycan 2) C1 TRUE
100038947 1 0.0026511 0.0026511 33.98330 0.8946906 0.1 0.3534567 LOC100038947 1593 protein_coding signal-regulatory protein beta 1-like C5 FALSE
14118 1 0.0050996 0.0050996 33.53175 0.8934236 0.1 0.3534567 Fbn1 10019 protein_coding fibrillin 1 C3 FALSE
17105 1 0.0003392 0.0003392 32.93845 0.8917118 0.1 0.3534567 Lyz2 1057 protein_coding lysozyme 2 C4 FALSE
67465 1 0.0002480 0.0002480 32.74887 0.8911531 0.1 0.3534567 Sf3a1 4916 protein_coding splicing factor 3a, subunit 1 C1 FALSE
18600 1 0.0016279 0.0016279 32.64502 0.8908446 0.1 0.3534567 Padi2 4850 protein_coding peptidyl arginine deiminase, type II C2 TRUE
237256 1 0.0013029 0.0013029 31.28856 0.8866488 0.1 0.3534567 Zc3h12d 4262 protein_coding zinc finger CCCH type containing 12D C5 TRUE
58217 1 0.0085137 0.0085137 29.31514 0.8799345 0.1 0.3534567 Trem1 3468 protein_coding triggering receptor expressed on myeloid cells 1 C3 TRUE

You can filter out the genes based on the $$R^2$$ value (the percentage variance explained by groups). There are 372 genes with $$R^2 \ge 0.7$$ which constitutes a fraction of 0.0397606 of all the genes in the dataset.

Here we print out 20 of the genes with highest $$R^2$$ value and the pvalue equal 0.1. P-value of 0.1 is the minimum possible when performing permutation test on distances between 6 observations split evenly into 2 groups (0.1 = 2/(6 choose 3)).

##  [1] "26374"     "320782"    "104156"    "27355"     "16763"     "13039"     "17394"     "66455"     "240913"    "219144"    "15370"     "213956"    "15530"     "100038947" "14118"     "17105"
## [17] "67465"     "18600"     "237256"    "58217"
## geom_smooth() using method = 'loess' and formula 'y ~ x'

Note, that in this analysis of variance approach we can specify a distance metric. Additionally, if you are interested in finding out genes DE in specific time intervals e.g. beginning or end of an experiment, you can choose to keep only the timepoints of interest are remove the rest from the time-courses stored in cop1.te@timecourse.data\$tc. trajectory_de() would then compute the distance for the specified time period and return DE genes at these specific time intervals.

# Functional pathways

In this section we will describe procedures to find functional pathways/gene sets corresponding to genes exhibiting differential expression profiles. We use publicly available reference databases to find relevant pathways.

You can select any list of genes for this step. However, the method is intended for sets of genes found differentially expressed using methods provided by TimeSeriesExperiment, discussed in the previous section.

Below, we use genes with DE trajectory, selected in the previous section. There are 372 in the set:

## [1] 372

Multiple functional pathways might be affected by knocking-out Cop1 gene. Therefore, we expect that genes found differential expressed can be parts of distinct pathways. Selected DE genes can be grouped by their cluster membership found earlier.

Below we plot expression profiles of selected DE genes, separated according to their cluster assignment.

We can test sets (clusters) of DE genes individually for pathway enrichment using pathwayEnrichment() which is built on top of goanna() and kegga() functions from limma package. The features argument provided to pathwayEnrichment() must be Entrez Gene IDs. The user must also specify the relevant species. Optionally, the user can modify the filtering parameters applied to enrichment results: fltr_DE, fltr_N, and fltr_P.DE. For details see the documentation ?pathwayEnrichment.

We can plot enrichment results for genes with differential trajectory in each cluster separately using plotEnrichment(). Here we plot the top n_max = 15 terms for each cluster. Next to enrichment terms, in the brackets, we have included information on the exact number of DE genes in a set (DE) and the size of the reference enrichment term (N). The points are colored by this fraction. The size of the points correspond the enrichment term size (N).

Below we print we print for example a plot for cluster C4 and C5