ENT3C is a method for qunatifying the similarity of micro-C/Hi-C derived chromosomal contact matrices. It is based on the von Neumann entropy¹ and recent work on entropy quantification of Pearson correlation matrices². For a contact matrix, ENT3C records the change in local pattern complexity of smaller Pearson-transformed submatrices along a matrix diagonal to generate a characteristic signal. Similarity is defined as the Pearson correlation between the respective entropy signals of two contact matrices.

https://doi.org/10.1093/nargab/lqae076

1. Loads cooler files and looks for shared empty bins.

2. ENT3C will first take the logarithm of an input matrix $\mathbf{M}$

3. Next, smaller submatrices $\mathbf{a}$ of dimension $n\times n$ are extracted along the diagonal of an input contact matrix $\mathbf{M}$

4. $nan$ values in $\mathbf{a}$ are set to the minimum value in $\mathbf{a}$.

5. $\mathbf{a}$ is transformed into a Pearson correlation matrix $\mathbf{P}$.

6. $\mathbf{P}$ is transformed into $\boldsymbol{\rho}=\mathbf{P}/n$ to fulfill the conditions for computing the von Neumann entropy.

7. The von Neumann entropy of $\boldsymbol{\rho}$ is computed as

   $S(\boldsymbol{\rho})=\sum_j \lambda_j \log \lambda_j$

   where $\lambda_j$ is the $j$ th eigenvalue of $\boldsymbol{\rho}$

8. This is repeated for subsequent submatrices along the diagonal of the input matrix and stored in the entropy signal $\mathbf{S}_{M}$.

9. Similarity $Q$ is defined as the Pearson correlation $r$ between the entropy signals of two matrices: $Q(\mathbf{M}_1,\mathbf{M}_2) = r(\mathbf{S}_{\mathbf{M}_1},\mathbf{S}_{\mathbf{M}_2})$.

Exemplary epiction of ENT3C derivation of the entropy signal $\mathbf{S}$ of two contact matrices $\mathbf{M}_1$ and $\mathbf{M}_2$. ENT3C's was run with submatrix dimension $n=300$, window shift $\varphi=10$, and maximum number of data points in $\boldsymbol{S}$, $\Phi_{\max}=\infty$, resulting in $\Phi=147$ submatrices. For subsequent scaled Pearson-transformed submatrices, $\boldsymbol{\rho}_i$, along the diagonal of $\log{\boldsymbol{M}}$, ENT3C computes the von Neumann entropies $S(\boldsymbol{\rho}_1), S(\boldsymbol{\rho}_2), \ldots, S(\boldsymbol{\rho}_{\Phi})$. The resulting signal $\mathbf{S} = \langle S(\boldsymbol{\rho}_{1}), S(\boldsymbol{\rho}_{2}), \ldots, S(\boldsymbol{\rho}_{\Phi}) \rangle$ is shown in blue under the matrix. The first two ($\boldsymbol{\rho}_{1-2}$), middle ($\boldsymbol{\rho}_{73}$), and last two submatrices ($\boldsymbol{\rho}_{146-147}$) are shown.

• https://pypi.org/project/ENT3C/

1. generate and activate python environment

   python3.11 -m venv .ent3c_venv
   
   source .ent3c_venv/bin/activate
   

2. install ENT3C:

   pip install ENT3C
   

Pre-built Linux executable is available in the Releases section (v.2.2.0).

• Download and make file executable.

chmod +x ./path/to/exe/ENT3C_exe

• For global use add path to ./bashrc file:

export PATH="$PATH:/path/to/exe/"

• 💡 note that Python or executable is recommended.

• compare_groups is not currently available for MATLAB and Julia implementation.

• Matlab scripts in matlab_version_ENT3C directory.

• Julia scripts in julia_version_ENT3C:

  • packages: DataFrames, BenchmarkTools, JSON, Printf, Plots, ColorSchemes, SuiteSparse, HDF5, NaNStatistics, Statistics, Combinatorics, CSV
  • For the Julia implementation, ubuntu's hdf5-tools is also required
  • Initial julia set-up
    • option for automatic global installation --install-deps=yes. (Works with any julia version)
    • predefined julia enviornments for julia versions 1.10.4 or 1.11.2 are defined in project_files/<v.v.v>/Manifest.toml and project_files/<v.v.v>/Project.toml
    • option to load enviornments with --resolve-env=yes and --julia-version=<v.v.v>

• CLI (python) usage:

   Usage:
   	ENT3C <command> --config=<path/to/config.json> [options]
  
   	Commands:
           get_entropy        Generates entropy output file <entropy_out_FN> .
           get_similarity            Generates similarity output file <similarity_out_FN> from <entropy_out_FN>.
           run_all            Generates <entropy_out_FN> and <similarity_out_FN>.
           compare_groups     Compare signal groups (requires --group1 and --group2 options)
  
   	Global Options:
           --config=<path>    Path to config JSON file (required for all commands)
  
   	<compare_groups> Options:
       	--group1=<GROUP>        First group name, must correspond to what comes before _BR* in config file.
       	--group2=<GROUP>        Second group name, must correspond to what comes before _BR* in config file.
  
   	Examples:
           ENT3C run_all --config=configs/myconfig.json
           ENT3C get_entropy --config=configs/myconfig.json
           ENT3C get_similarity --config=configs/myconfig.json
           ENT3C compare_groups --config=configs/myconfig.json --group1=H1-hESC --group2=K562
  

• For linux executable use: ENT3C_exe <command> --config=<path/to/config.json> [options]

• alternatively run ENT3C in python as:

   import ENT3C
  
   ENT3C_OUT = ENT3C.run_get_entropy("config/myconfig.json")
  
   Similarity = ENT3C.run_get_similarity("config/myconfig.json")
  
   ENT3C_OUT, Similarity = ENT3C.run_all("config/myconfig.json")
  
   EUCLIDEAN = ENT3C.run_compare_groups("config/myconfig.json",group1,group2)
  
  

• initial call for global package installation (see "initial julia set-up"):

   julia ENT3C.jl --config-file=config/config.test.json --install-deps=yes
  

• after initialization:

   julia ENT3C.jl --config-file=config/config.json
  

• alternativly load the predefined enviornments for julia 1.10.4 or 1.11.2

   julia ENT3C.jl --config-file=config/config.json --resolve-env=yes --julia-version=<v.v.v>
  

💡 note the matlab and julia implementations will always generate both the entropy and similarity dataframes

matlab -nodesktop -nosplash -nodisplay -r "ENT3C('config/config.json'); exit"

💡 note the matlab and julia implementations will always generate the entropy and similarity dataframes

• all ENT3C parameters are defined in .json files config/config.json. Examples can be found in config directory.

• Paremeters defined in <config_file>:

  1. The main ENT3C parameter affecting the final entropy signal $S$ is the dimension of the submatrices SUB_M_SIZE_FIX.

     • "SUB_M_SIZE_FIX": <integer> $\dots$ fixed submatrix dimension.

       • SUB_M_SIZE_FIX can be either be fixed by or alternatively, one can specify CHRSPLIT; in this case SUB_M_SIZE_FIX will be computed internally to fit the number of desired times the contact matrix is to be paritioned into.

       PHI=1+floor((N-SUB_M_SIZE)./phi)

       where N is the size of the input contact matrix, phi is the window shift, PHI is the number of evaluated submatrices (consequently the number of data points in $S$).

     • "CHRSPLIT": <integer> $\dots$ number of times into which a $N \times N$ contact matrix is partitioned into which defining SUB_M_SIZE_FIX = floor(N/CHRSPLIT+0.5). If specified, then "SUB_M_SIZE_FIX": null otherwise "CHRSPLIT": null.

  2. "DATA_PATH": </path/to/data> $\dots$ input data path.

  3. input files in format: [<COOL_FILENAME>, <SHORT_NAME>]

     "FILES": [
     	"ENCSR079VIJ.BioRep1.40kb.cool",
     	"G401_BR1",
     	"ENCSR079VIJ.BioRep2.40kb.cool",
     	"G401_BR2"]
     

     • Any biological replicates must be indicated in <SHORT_NAME> using the suffix "_BR%d".

     • Note: ENT3C also takes mcool files as input.

  4. "`OUT_DIR": "<desired_output_directory_name>" $\dots$ output directory. OUT_DIR will be concatenated with OUTPUT/JULIA/ or OUTPUT/MATLAB/.

  5. "OUT_PREFIX": "<desired_output_prefix_>" $\dots$ prefix for output files.

  6. "Resolution": "<integer,integer,...>" e.g. "40e3,100e3" $\dots$ resolutions to be evaluated.

  7. "ChrNr": "<integer,integer,...>" "15,16,17,18,19,20,21,22,X" $\dots$ chromosome numbers to be evaluated.

  8. "NormM": <0|1> $\dots$ input contact matrices can be balanced. If NormM: 1, balancing weights in cooler are applied. If set to 1, ENT3C expects weights to be in dataset /resolutions/<resolution>/bins/<WEIGHTS_NAME>.

  9. "WEIGHTS_NAME": "<name_of_weights>" $\dots$ name of dataset in cooler containing normalization weights.

  10. "phi": <integer> $\dots$ number of bins to the next matrix.

  11. "PHI_MAX": <integer> $\dots$ number of submatrices; i.e. number of data points in entropy signal $S$. If set, $\varphi$ is increased until $\Phi \approx \Phi_{\max}$.

1. <OUT_DIR>/<OUTPUT_PREFIX>_ENT3C_similarity.csv $\dots$ will contain all combinations of comparisons. The second two columns contain the short names specified in FILES and the third column Q the corresponding similarity score. OUTPUT/PYTHON/EvenChromosomes_NoWeights_ENT3C_similarity.csv:

   Resolution	ChrNr	Sample1		Sample2		Q
   40000		2		G401_BR1	G401_BR2	0.9978330002118974
   40000		2		G401_BR1	LNCap_BR1	0.4129094106283695
   40000		2		G401_BR1	LNCap_BR2	0.3049196919642929
   .			.		.			.			.
   .			.		.			.			.
   .			.		.			.			.
   

2. <OUT_DIR>/<OUTPUT_PREFIX>_ENT3C_OUT.csv $\dots$ ENT3C output table. OUTPUT/PYTHON/EvenChromosomes_NoWeights_ENT3C_OUT.csv:

   Name		ChrNr	Resolution	n	PHI	phi	binNrStart	binNrEnd	START	END			S
   G401_BR1	2		40000		500	918	6	0			499			0		20000000	3.7896426915562462
   G401_BR1	2		40000		500	918	6	6			505			240000	20240000	3.789044181663418
   G401_BR1	2		40000		500	918	6	12			511			480000	20480000	3.7918253959272032
   .			.		.			.	.	.	.			.			.		.			.
   .			.		.			.	.	.	.			.			.		.			.
   .			.		.			.	.	.	.			.			.		.			.
   

   Each row corresponds to an evaluated submatrix with fields Name (the short name specified in FILES), ChrNr, Resolution, the sub-matrix dimension sub_m_dim, PHI=1+floor((N-SUB_M_SIZE)./phi), binNrStart and binNrEnd correspond to the start and end bin of the submatrix, START and END are the corresponding genomic coordinates and S is the computed von Neumann entropy.

   • Example of output generated for ENT3C get_entropy --config=config/myconfig.json:
     • EvenChromosomes_NoWeights_40kb_ENT3C_signals.pdf
     • unbalanced 40kb contact matrices for even chromosomes across 5 cell lines. SUB_MATRIX_SIZE was 500:

3. <OUT_DIR>/<OUTPUT_PREFIX>_Eucl_<group1>vs<group2>.csv $\dots$ Euclidean distance between average z-scores of S over <group1> and <group2>: (here group1=HFFc6, group2=G401). Arranged in descending order of $meanS_Euclidean$.

   Resolution	ChrNr	START		END			meanS_Euclidean
   40000		6		62360000	82360000	3.3625023926723685
   40000		6		62120000	82120000	3.3546076641065095
   40000		6		61880000	81880000	3.3441925121710026
   .			.		.			.			.
   .			.		.			.			.
   .			.		.			.			.
   

   • Example of first page of output generated for ENT3C compare_groups --config=config/myconfig.json --group1 = HFFc6 group2 = "G401"
     • EvenChromosomes_NoWeights_Eucl_40kb_HFFc6vsG401.pdf

Both Julia and MATLAB implementations (ENT3C.jl and ENT3C.m) were tested on Hi-C and micro-C contact matrices binned at 40 kb in cool format.

micro-C

Hi-C

1. for the Hi-C data, bam files were downloaded from the ENCODE data portal and converted into pairs files using the pairtools parse function³

   pairtools parse --chroms-path hg38.fa.sizes -o <OUT.pairs.gz> --assembly hg38 --no-flip --add-columns mapq --drop-sam --drop-seq --nproc-in 15 --nproc-out 15 <IN.bam>

2. for the micro-C data, pairs of technical replicates (TRs) were merged with pairtools merge. E.g. for H1-hESC, BR1 (4DNES21D8SP8):

   pairtools merge -o <hESC.BR1.pairs.gz> --nproc 10 4DNFING6ZFDF.pairs.gz 4DNFIBMG8YA3.pairs.gz 4DNFIMT4PHZ1.pairs.gz 4DNFI8GM4EL9.pairs.gz

3. 40 kb coolers were generated from the Hi-C/micro-C pairs files with cload pairs function⁴

   cooler cload pairs -c1 2 -p1 3 -c2 4 -p2 5 --assembly hg38 <CHRSIZE_FILE:40000> <IN.pairs.gz> <OUT.cool>

1. Neumann, J. von., Thermodynamik quantenmechanischer Gesamtheiten. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1927. 1927. 273-291.
2. Felippe, H., et. al., Threshold-free estimation of entropy from a pearson matrix. EPL. 141(3):31003. 2023.
3. Open2C et. al., Pairtools: from sequencing data to chromosome contacts. bioRxiv. 2023.
4. Abdennur,N., and Mirny, L.A., Cooler: scalable storage for Hi-C data and other genomically labeled arrays. Bioinformatics. 2020.