# SpectralTAD Vignette

Kellen Cresswell1, John Stansfield1 and Mikhail Dozmorov1

1Department of Biostatistics, Virginia Commonwealth University, Richmond, VA

#### May 3, 2019

#### Abstract

SpectralTAD is an R package designed to identify Topologically Associated Domains (TADs) from Hi-C contact matrices. It uses a modified version of spectral clustering that uses a sliding window to quickly detect TADs. The function works on a range of different formats of contact matrices and returns a list of data frames or GRanges in BED format with TAD coordinates, seperated by hierarchy level. The method does not require users to adjust any parameters to work and gives them control over the number of levels to be returned.

#### Package

SpectralTAD 1.0.0

# Contents

# 1 Introduction

`SpectralTAD` is a package designed to allow for fast hierarchical TAD calling on a wide-variety of chromatin conformation capture (Hi-C) data. `SpectralTAD` takes a contact matrix as an input and outputs a list of data frames or GRange objects in BED format containing coordinates corresponding to TAD locations. The package contains two functions, `SpectralTAD()` and `SpectralTAD_Par()` with SpectralTAD being a single-CPU TAD-caller and `SpectralTAD_Par` being the parallelized version. This package provides flexibility in the data it accepts, allowing for \(n \times n\) (square numerical matrix), \(n \times (n+3)\) (square numerical matrix with the additional chromosome, start, end columns), and 3-column sparse matrices (described below). There are no parameters required for running the functions, though certain methods for customizing the results are available. The idea is to give users the freedom to control how they call TADs while giving them the option to perform it in an unsupervised manner.

The package is based around the windowed spectral clustering algorithm, introduced by (Cresswell, Stansfield, and Dozmorov 2019) , which is designed to be robust to sparsity, noise, and sequencing depth of Hi-C data.

# 2 Getting Started

## 2.1 Installation

    # if (!requireNamespace("BiocManager", quietly=TRUE))
    #     install.packages("BiocManager")
    # BiocManager::install("SpectralTAD")
    devtools::install_github("cresswellkg/SpectralTAD")
    library(SpectralTAD)

## 2.2 Input data

### 2.2.1 Working with \(n \times n\) matrices

\(n \times n\) contact matrices, are most commonly associated with data coming from the Bing Ren lab (http://chromosome.sdsc.edu/mouse/hi-c/download.html). These contact matrices are square and symmetric with entry \(ij\) corresponding to the number of contacts between region \(i\) and region \(j\). Below is an example of a \(5 \times 5\) region of an \(n \times n\) contact matrix. Derived from (Rao et al. 2014), chromosome 20 data at 25kb resolution. Note the symmetry around the diagonal - the typical shape of chromatin interaction matrix.

    ##             50000    75000   100000   125000   150000
    ##    50000     2021      727      203      205      155
    ##    75000      727     3701      583      491      348
    ##   100000      203      583     2023      603      274
    ##   125000      205      491      603     3350      676
    ##   150000      155      348      274      676     2528

### 2.2.2 Working with \(n \times (n+3)\) matrices

\(n \times (n+3)\) matrices are commonly associated with the TopDom tad-caller (http://zhoulab.usc.edu/TopDom/). These matrices consist of a normal \(n \times n\) matrix but with 3 additional leading columns containg the chromosome, the start of the region and the end of the region. Regions in this case are determined by the resolution of the data. The typical \(n \times (n+3)\) matrix is shown below.

    ##
    ## 1  chr19  50000  75000 2021  727  203  205  155  165  193
    ## 2  chr19  75000 100000  727 3701  583  491  348  335  350
    ## 3  chr19 100000 125000  203  583 2023  603  274  228  213
    ## 4  chr19 125000 150000  205  491  603 3350  676  447  390
    ## 5  chr19 150000 175000  155  348  274  676 2528  732  463
    ## 6  chr19 175000 200000  165  335  228  447  732 3497  998
    ## 7  chr19 200000 225000  193  350  213  390  463  998 4450
    ## 8  chr19 225000 250000  135  236  137  259  301  543 1061
    ## 9  chr19 250000 275000  158  259  183  299  317  554  731
    ## 10 chr19 275000 300000   96  166  115  171  179  319  416

### 2.2.3 Working with sparse 3-column matrices

Sparse 3-column matrices, sometimes referred to as a coordinated lists, are matrices where the first and second column refer to region \(i\) and region \(j\) of the chromosome, and the third column is the number of contacts between them. This style is becoming increasingly popular and is associated with raw data from Rao (https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE63525), and is the data output produced by the Juicer tool (Durand et al. 2016). 3-column matrices are handled internally in the package by converting them to \(n \times n\) matrices using the `HiCcompare` package’s `sparse2full()` function. The first 5 rows of a typical sparse 3-column matrix is shown below.

    ##    region1 region2   IF
    ## 1:   50000   50000 2021
    ## 2:   50000   75000  727
    ## 3:   75000   75000 3701
    ## 4:   50000  100000  203
    ## 5:   75000  100000  583

### 2.2.4 Working with other data types

Users can also find TADs from data output by `cooler` (http://cooler.readthedocs.io/en/latest/index.html) and HiC-Pro (https://github.com/nservant/HiC-Pro) with minor pre-processing using the `HiCcompare` package.

#### 2.2.4.1 Working with .cool files

The cooler software can be downloaded from http://cooler.readthedocs.io/en/latest/index.html. It essentially provides access to a catalog of popular HiC datasets. We can pre-process and use .cool files that are associated with cooler files using the following steps:

1. Download `.cool` file from (ftp://cooler.csail.mit.edu/coolers)
2. Convert to text file using `cooler dump --join Rao2014-GM12878-DpnII-allreps-filtered.50kb.cool > Rao.GM12878.50kb.txt`
3. Run the code below

    #Read in data
    cool_mat = read.table("Rao.GM12878.50kb.txt")
    #Convert to sparse 3-column matrix using cooler2sparse from HiCcompare
    sparse_mats = HiCcompare::cooler2sparse(cool_mat)
    #Remove empty matrices if necessary
    #sparse_mats = sparse_mats$cis[sapply(sparse_mats, nrow) != 0]
    #Run SpectralTAD
    spec_tads = lapply(names(sparse_mats), function(x) {
      SpectralTAD(sparse_mats[[x]], chr = x)
    })

#### 2.2.4.2 Working with HiC-Pro files

HiC-Pro data comes with 2 files, the `.matrix` file and the `.bed` file. The `.matrix` file is a 3-column matrix where instead of coordinates as the 1st and 2nd column, there is an ID. The `.bed` file maps these IDs to genomic coordinates. The steps for analyzing these files is shown below:

    #Read in both files
    mat = read.table("amyg_100000.matrix")
    bed = read.table("amyg_100000_abs.bed")
    #Convert to modified bed format
    sparse_mats = HiCcompare::hicpro2bedpe(mat,bed)
    #Remove empty matrices if necessary
    #sparse_mats$cis = sparse_mats$cis[sapply(sparse_mats, nrow) != 0]
    #Go through all matrices
    sparse_tads = lapply(sparse_mats$cis, function(x) {
      #Pull out chromosome
      chr = x[,1][1]
      #Subset to make three column matrix
      x = x[,c(2,5,7)]
      #Run SpectralTAD
      SpectralTAD(x, chr=chr)
    })

## 2.3 Running SpectralTAD

Once matrices are in an acceptable format, SpectralTAD can be run with as little as two parameters or as many as eight. Below we show how to run the algorithm with just TAD detection and no quality filtering.

    #Get the rao contact matrix built into the package
    data("rao_chr20_25_rep")
    head(rao_chr20_25_rep)

    ##       V1     V2   V3
    ## 1  50000  50000 2021
    ## 2  50000  75000  727
    ## 3  75000  75000 3701
    ## 4  50000 100000  203
    ## 5  75000 100000  583
    ## 6 100000 100000 2023

    #We see that this is a sparse 3-column contact matrix
    #Running the algorithm with resolution specified
    results = SpectralTAD(rao_chr20_25_rep, chr = "chr20", resolution = 25000, qual_filter = FALSE, z_clust = FALSE)
    #Printing the top 5 TADs
    head(results$Level_1, 5)

    ## # A tibble: 5 x 4
    ##   chr     start     end Level
    ##   <chr>   <dbl>   <dbl> <dbl>
    ## 1 chr20   50000 1200000     1
    ## 2 chr20 1200000 2450000     1
    ## 3 chr20 2450000 3525000     1
    ## 4 chr20 3525000 4075000     1
    ## 5 chr20 4075000 4575000     1

    #Repeating without specifying resolution
    no_res = SpectralTAD(rao_chr20_25_rep, chr = "chr20", qual_filter = FALSE, z_clust = FALSE)
    #We can see below that resolution can be estimated automatically if necessary
    identical(results, no_res)

    ## [1] TRUE

## 2.4 Filtering TADs

### 2.4.1 Silhouette score filtering

One method for filtering TADs is using group-wise silhouette scores. This is done by getting the overall silhouette for each group and removing those with a score of less than .25. A low silhouette score for a given TAD indicates a poor level of connectivity within it.

    #Running SpectralTAD with silhouette score filtering
    qual_filt = SpectralTAD(rao_chr20_25_rep, chr = "chr20", qual_filter = TRUE, z_clust = FALSE, resolution = 25000)
    #Showing quality filtered results
    head(qual_filt$Level_1,5)

    ## # A tibble: 5 x 5
    ##   chr     start     end Sil_Score Level
    ##   <chr>   <dbl>   <dbl>     <dbl> <dbl>
    ## 1 chr20   50000 1200000     0.608     1
    ## 2 chr20 1200000 2450000     0.682     1
    ## 3 chr20 2450000 3525000     0.450     1
    ## 4 chr20 3525000 4075000     0.758     1
    ## 5 chr20 4075000 4575000     0.668     1

    #Quality filtering generally has different dimensions
    dim(qual_filt$Level_1)

    ## [1] 61  5

    dim(results$Level_1)

    ## [1] 62  4

The results when using the quality filtering option are altered to include a new column called `Sil_Score`. This column includes a group-wise silhouette score. In this example a single TAD was removed due to having poor organization (Low silhouette score).

### 2.4.2 Z-score filtering

Z-score filtering is based on observations about the log-normality of the distance between eigenvector gaps from (Cresswell, Stansfield, and Dozmorov 2019). Z-score filtering bypasses the uses of silhouette score and defines a TAD boundary as the area between consecutive regions where the z-score of the eigenvector gap is greater than 2.

    z_filt = SpectralTAD(rao_chr20_25_rep, chr = "chr20", qual_filter = FALSE, z_clust = TRUE, resolution = 25000)
    head(z_filt$Level_1, 5)

    ## # A tibble: 5 x 4
    ##   chr     start     end Level
    ##   <chr>   <dbl>   <dbl> <dbl>
    ## 1 chr20   50000  775000     1
    ## 2 chr20  775000 1225000     1
    ## 3 chr20 1225000 2200000     1
    ## 4 chr20 2200000 2475000     1
    ## 5 chr20 2475000 3050000     1

    dim(z_filt$Level_1)

    ## [1] 85  4

We can see that TADs found using this method are generally more fine than those found using the silhouette score based filtering. Z-score filtering is more suited for people not interested in TAD hierarchies because the initial TADs detected are less broad than those by quality filtering.

## 2.5 Finding hierarchical TADS

Our method is specifically designed to find hierarchical TADs. To do so, users must specify how many levels they are interested in using the `levels` parameter. There is no limit to the number of levels but after a certain point no new TADs will be found due to limitations in TAD size or TAD quality. Hierarchies are found by running the algorithm initially and then iteratively running it on sub-TADs until none can be found. At the levels below the initial level we use z-score filtering by default.

    #Running SpectralTAD with 3 levels and no quality filtering
    spec_hier = SpectralTAD(rao_chr20_25_rep, chr = "chr20", resolution = 25000, qual_filter = FALSE, levels = 3)
    #Level 1 remains unchanged
    head(spec_hier$Level_1,5)

    ## # A tibble: 5 x 4
    ##   chr     start     end Level
    ##   <chr>   <dbl>   <dbl> <dbl>
    ## 1 chr20   50000  775000     1
    ## 2 chr20  775000 1225000     1
    ## 3 chr20 1225000 2200000     1
    ## 4 chr20 2200000 2475000     1
    ## 5 chr20 2475000 3050000     1

    #Level 2 contains the sub-TADs for level 1
    head(spec_hier$Level_2,5)

    ## # A tibble: 5 x 4
    ##   chr     start     end Level
    ##   <chr>   <dbl>   <dbl> <dbl>
    ## 1 chr20   50000  350000     2
    ## 2 chr20  350000  775000     2
    ## 3 chr20 1225000 1750000     2
    ## 4 chr20 1750000 2200000     2
    ## 5 chr20 2475000 2700000     2

    #Level 3 contains even finer sub-TADs for level 1 and level 2
    head(spec_hier$Level_3,5)

    ## # A tibble: 5 x 4
    ##   chr     start     end Level
    ##   <chr>   <dbl>   <dbl> <dbl>
    ## 1 chr20  350000  550000     3
    ## 2 chr20  550000  775000     3
    ## 3 chr20 1225000 1475000     3
    ## 4 chr20 1475000 1750000     3
    ## 5 chr20 1750000 1975000     3

Note that as we move down levels, more gaps form indicating regions where sub-TADs are not present.

## 2.6 Removing gaps

Though, it has been shown that windowed spectral clustering is more robust to sparsity than other common methods(Cresswell, Stansfield, and Dozmorov 2019), there is still some instability caused by consecutive regions of gaps. To counter this, we use a `gap_threshold` parameter to allow users to exclude regions based on how many zeros are included. By default this value is set to 1 which means only columns/rows with 100% of values being zero are removed before analysis. Accordingly, setting this value to .7 would mean rows/columns with 70% zeros would be removed. Since we are not interested in long-range contacts for TAD identification, this percentage only applies to the number of zeros within a specific distance of the diagonal (Defined by the maximum TAD size). Users must be careful not to filter too much as this can remove informative regions of the matrix.

## 2.7 Running SpectralTAD with parallelization

It is sometimes the case that people want to run SpectralTAD on multiple chromosomes at once in parallel. This can be done using the `SpectralTAD_Par` function. SpectralTAD_Par is identical to SpectralTAD but takes a list of contact matrices as an input. These matrices can be of any of the allowable types and mixing of types is allowed. Users are required to provide a vector of chromosomes, `chr_over`, where it is ordered such that each entry matches up with its corresponding contact matrix in the list of matrices. Users can also input list names using the `labels` parameter. In terms of parallelization, users can either input the number of cores they would like to use or the function will automatically use 1 less than the total number of cores on the computer on which it is ran. The function works on Linux/Mac and Windows with automatic OS detection built in. We show the steps below.

    #Creating replicates of our HiC data for demonstration
    cont_list = replicate(3,rao_chr20_25_rep, simplify = FALSE)
    #Creating a vector of chromosomes
    chr_over = c("chr20", "chr20", "chr20")
    #Creating a list of labels
    labels = c("Replicate 1", "Replicate 2", "Replicate 3")
    SpectralTAD_Par(cont_list = cont_list, chr_over = chr_over, labels = labels, cores = 3)

## 2.8 Effect of matrix type on runtime

The type of matrix input into the algorithm can affect runtimes for the algorithm. \(n \times n\) matrices require no conversion and are the fastest. Meanwhile, \(n \times (n+3)\) matrices take slightly longer to run due to the need to remove the first 3 columns. Sparse 3-column matrices have the highest runtimes due to the complexity of converting them to an \(n \times n\) matrix. The times are summarized below, holding all other parameters constant.

    library(microbenchmark)
    #Converting to nxn
    n_n = HiCcompare::sparse2full(rao_chr20_25_rep)
    #Converting to nxn+3
    n_n_3 = cbind.data.frame("chr20", as.numeric(colnames(n_n)), as.numeric(colnames(n_n))+25000, n_n)
    #Defining each function
    sparse = SpectralTAD(cont_mat = rao_chr20_25_rep, chr = "chr20", qual_filter = FALSE)
    n_by_n = SpectralTAD(cont_mat = n_n, chr = "chr20", qual_filter = FALSE)
    n_by_n_3 =SpectralTAD(cont_mat = n_n_3, chr = "chr20", qual_filter = FALSE)

    #Benchmarking different parameters
    microbenchmark(sparse = SpectralTAD(cont_mat = rao_chr20_25_rep, chr = "chr20", qual_filter = FALSE),
    n_by_n = SpectralTAD(cont_mat = n_n, chr = "chr20", qual_filter = FALSE),
    n_by_n_3 =SpectralTAD(cont_mat = n_n_3, chr = "chr20", qual_filter = FALSE), unit = "s", times = 3)

    ## Unit: seconds
    ##      expr       min        lq      mean    median        uq       max neval cld
    ##    sparse 0.8315568 1.0594499 1.1493487 1.2873431 1.3082447 1.3291463     3   b
    ##    n_by_n 0.2993548 0.2998657 0.4874858 0.3003766 0.5815512 0.8627259     3  a
    ##  n_by_n_3 0.3725899 0.3729177 0.3913872 0.3732455 0.4007859 0.4283262     3  a

## 2.9 Effect of parameters on runtime

Just as the type of data affects runtime, the parameters used to detect TADs do as well. The main bottleneck is the quality filter function which requires the inversion of a matrix and the calculation of silhouette score.

    microbenchmark(quality_filter = SpectralTAD(cont_mat = n_n, chr = "chr20", qual_filter = TRUE, z_clust = FALSE), no_filter = SpectralTAD(cont_mat = n_n, chr = "chr20", qual_filter = FALSE, z_clust = FALSE), z_clust = SpectralTAD(cont_mat = n_n, chr = "chr20", qual_filter = FALSE, z_clust = TRUE), times = 3, unit = "s")

    ## Unit: seconds
    ##            expr       min        lq      mean    median        uq       max neval cld
    ##  quality_filter 1.1006827 1.1128617 1.1202709 1.1250407 1.1300650 1.1350892     3   c
    ##       no_filter 0.8292907 0.8352378 0.8652519 0.8411850 0.8832326 0.9252802     3  b
    ##         z_clust 0.2842923 0.2887196 0.2928393 0.2931469 0.2971129 0.3010788     3 a

As can be seen using the z-score method is the fastest.

# References

Cresswell, Kellen G, John C Stansfield, and Mikhail G Dozmorov. 2019. “SpectralTAD: An R Package for Defining a Hierarchy of Topologically Associated Domains Using Spectral Clustering,” February. Cold Spring Harbor Laboratory. https://doi.org/10.1101/549170.

Durand, Neva C., Muhammad S. Shamim, Ido Machol, Suhas S.P. Rao, Miriam H. Huntley, Eric S. Lander, and Erez Lieberman Aiden. 2016. “Juicer Provides a One-Click System for Analyzing Loop-Resolution Hi-c Experiments.” Cell Systems 3 (1). Elsevier BV:95–98. https://doi.org/10.1016/j.cels.2016.07.002.

Rao, Suhas S.P., Miriam H. Huntley, Neva C. Durand, Elena K. Stamenova, Ivan D. Bochkov, James T. Robinson, Adrian L. Sanborn, et al. 2014. “A 3D Map of the Human Genome at Kilobase Resolution Reveals Principles of Chromatin Looping.” Cell 159 (7). Elsevier BV:1665–80. https://doi.org/10.1016/j.cell.2014.11.021.