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/10.1101/2024.01.30.577923

Summary of ENT3C approach

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

2. ENT3C will extract smaller submatrices $\mathbf{a}$ of dimension $n\times n$ along the diagonal of an input contact matrix $\mathbf{M}$

3. The logarithm of $\mathbf{a}$ is taken ($nan$ values are set to zero).

4. $\mathbf{a}$ is transformed into a Pearson correlation matrix $\mathbf{P}$ ($nan$ values are set to zero).

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

6. 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 $\boldsymbol{\rho}$

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

8. 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 $WS=10$, and maximum number of data points in $\boldsymbol{S}$, $WN_{\max}=\infty$, resulting in $WN=147$ submatrices. For subsequent scaled Pearson-transformed submatrices, $\boldsymbol{\rho}_i$, along the diagonal of $\log{\boldsymbol{A}}$, ENT3C computes the von Neumann entropies $S(\boldsymbol{\rho}_1), S(\boldsymbol{\rho}_2), \ldots, S(\boldsymbol{\rho}_{WN})$. The resulting signal $\mathbf{S} = \langle S(\boldsymbol{\rho}_{1}), S(\boldsymbol{\rho}_{2}), \ldots, S(\boldsymbol{\rho}_{WN}) \rangle$ 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.

Requirements

Julia or MATLAB

Data

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

Cell line	Biological Replicate (BR)	Accession (Experiemnt set)	Accession (pairs)
H1-hESC	1	4DNES21D8SP8	4DNFING6ZFD, 4DNFIBMG8YA3, 4DNFIMT4PHZ1, 4DNFI8GM4EL9
H1-hESC	2	4DNES21D8SP8	4DNFIIYUGYBU, 4DNFI89L17XY, 4DNFIXP9MVBU, 4DNFI2YHYAJO, 4DNFIULY29IQ
HFFc6	1	4DNESWST3UBH	4DNFIN7IIIY6, 4DNFIJZDEIZ3, 4DNFIYBTHGNA, 4DNFIK8UIB5B
HFFc6	2	4DNESWST3UBH	4DNFIF5F4HRG, 4DNFIK82YRNM, 4DNFIATCW955, 4DNFIZU6ADT1, 4DNFIKWV6BY2
HFFc6	3	4DNESWST3UBH	4DNFIFJL4JIH, 4DNFIONHB78N, 4DNFIG1ZOVIM, 4DNFIPKVL9YI, 4DNFIJM966UR, 4DNFIV8JNJB8

Hi-C

Cell line	Biological Replicate (BR)	Accession (Experiemnt set)	Accession (BAM)
G401	1	ENCSR079VIJ	ENCFF649MAY
G401	2	ENCSR079VIJ	ENCFF758WUD
LNCaP	1	ENCSR346DCU	ENCFF977XHB
LNCaP	2	ENCSR346DCU	ENCFF204XII
A549	1	ENCSR444WCZ	ENCFF867DCM
A549	2	ENCSR444WCZ	ENCFF532XBC

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>

Configuration File

Both Julia and MATLAB implementations (ENT3C.jl and ENT3C.m) call a configuration file in JSON format.

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

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.

WN=1+floor((N-SUB_M_SIZE)./WS)

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

ENT3C parameters set in config/config.json

DATA_PATH: "DATA" $\dots$ input data path.

FILES: ["ENCSR079VIJ.BioRep1.40kb.cool","G401_BR1" ...] $\dots$ input files in format: [<COOL_FILENAME>, <SHORT_NAME>] 💡 ENT3C also takes mcool files as input. Please refer to biological replicates as "_BR%d" in the <SHORT_NAME>.

OUT_DIR: "OUTPUT/" $\dots$ output directory. OUT_DIR will be concatenated with OUTPUT/JULIA/ or OUTPUT/MATLAB/.

OUT_PREFIX: "40kb" $\dots$ prefix for output files.

Resolution: 40000 $\dots$ resolution to be evaluated.

ChrNr: 14 $\dots$ chromosome number to be evaluated.

NormM: 0 $\dots$ input contact matrices can be balanced. If NormM:1, balancing weights in cooler are applied.

SUB_M_SIZE_FIX: null $\dots$ fixed submatrix dimension.

CHRSPLIT: 10 $\dots$ number of submatrices into which the contact matrix is partitioned into.

WS: 1 $\dots$ number of bins to the next matrix.

WN_MAX: 1000 $\dots$ number of submatrices; i.e. number of data points in entropy signal $S$. If set, $WS$ is increased until $WN \approx WN_{\max}$.

Running main scripts

Upon modifying config/config.json as desired, ENT3C.jl and ENT3C.m will run with using specified parameters.

Associated functions are contained in directories JULIA_functions/ and MATLAB_functions/.

Output files: 40kb_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.

ChrNr	Sample1	Sample2	Q
15	A549_BR1	A549_BR2	0.991055440056682
15	A549_BR1	G401_BR1	0.558213759737198
15	A549_BR1	G401_BR2	0.593612526651996
.		.		.
.		.		.
.		.		.

40kb_ENT3C_OUT.csv $\dots$ ENT3C output table.

Name	ChrNr	Resolution	sub_m_dim	WN	WS	binNrStart	binNrEND	START	END	S
G401_BR1	15	40000	205	1841	1	1	282	0	11280000	3.27338758846104
G401_BR1	15	40000	205	1841	1	2	283	40000	11320000	3.27061968317512
G401_BR1	15	40000	205	1841	1	3	284	80000	11360000	3.25483843736616
.		.	.		.	.	.		.		.		.		.
.		.	.		.	.	.		.		.		.		.
.		.	.		.	.	.		.		.		.		.

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, WN=1+floor((N-SUB_M_SIZE)./WS), 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.

40kb_ENT3C_signals.png $\dots$ simple visualization of entropy signals $S$:

Entropy signals $S$ generated by the Julia script ENT3C.jl for contact matrices of chromosome 15-22 binned at 40 kb in various cell lines:

Entropy signals $S$ generated by the MATLAB script ENT3C.m for contact matrices of chromosomes 15-22 binned at 40 kb in various cell lines.

References

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.