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# MIA-Sig: multiplex chromatin interaction analysis by signal processing and statistical algorithms

*Genome Biology*
**volume 20**, Article number: 251 (2019)

## Abstract

The single-molecule multiplex chromatin interaction data are generated by emerging 3D genome mapping technologies such as GAM, SPRITE, and ChIA-Drop. These datasets provide insights into high-dimensional chromatin organization, yet introduce new computational challenges. Thus, we developed MIA-Sig, an algorithmic solution based on signal processing and information theory. We demonstrate its ability to de-noise the multiplex data, assess the statistical significance of chromatin complexes, and identify topological domains and frequent inter-domain contacts. On chromatin immunoprecipitation (ChIP)-enriched data, MIA-Sig can clearly distinguish the protein-associated interactions from the non-specific topological domains. Together, MIA-Sig represents a novel algorithmic framework for multiplex chromatin interaction analysis.

## Background

Traditional 3D genome mapping efforts have suggested complex chromosomal folding structures. In particular, methods based on high-throughput sequencing capture bulk chromatin contacts (Hi-C; Lieberman-Aiden et al. [18]) or enrich for chromatin contacts involving a specific protein (ChIA-PET; Fullwood et al. [9]). Both of these methods rely on proximity ligation and therefore can only reveal population averages of pairwise contacts. Thus, they lacked the ability to simultaneously capture multiple loci involved in a chromatin complex in an individual cell.

To overcome these drawbacks, novel experimental methods have recently been developed to capture multiplex chromatin contacts with single-molecule resolution. For instance, GAM (Beagrie et al. [2]) identifies multi-way interactions by capturing multiple DNA elements co-existing in a given nuclear slice, SPRITE (Quinodoz et al. [25]) barcodes individual chromatin complexes via a split-pool strategy, and ChIA-Drop (Zheng et al. [31]) partitions each complex into a microfluidic droplet for barcoding and amplification. Collectively, these emerging 3D genome mapping technologies are advancing the frontier of the nuclear architecture field. However, as with other genomic approaches prone to the background noise, the noisy and high-dimensional nature of the multiplex data poses unique computational challenges that cannot be readily addressed with existing tools that are tailored for pairwise interactions data.

Numerous software tools are available for analyzing data generated by
genome-wide 3D architecture assays such as 3C, 4C, 5C, and the most common assay
Hi-C. For example, HiCNorm (Hu et al. [10]) and Hi-Corrector (Li et al. [17]) explicitly or implicitly correct the bias observed in Hi-C
data. Fit-Hi-C (Ay et al. [1]) and
GOTHiC (Mifsud et al. [20]) aim to
assess the statistical significance of intra-chromosomal contacts by incorporating
bias in the background null model. The authors of Fit-Hi-C emphasized the importance
of accurately modeling the inverse relationship between genomic distance and contact
probability. Similarly, multiplex data also depend on the distance, but currently
available tools cannot be naively applied since (1) genomic distance is now
multi-dimensional instead of 1D, i.e., a complex with *n*-way contacts yield *n* − 1
neighboring distances and \( \left(\genfrac{}{}{0pt}{}{n}{2}\right)=\frac{n\left(n-1\right)}{2} \) pairwise distances, and (2) contact probability must be defined
for all *n*-way contacts, yet it is unclear if
ten-way contact is as likely as two-way contact.

Another crucial component in Hi-C data analysis is to call topologically
associating domains (TADs), loosely defined as regions with more contacts inside
than outside. In general, TADs appear as squares along the diagonal in the contact
map, and the goal is to identify and segment the genome. There are more than 20 TAD
calling algorithms (Zufferey et al. [32]), some of which convert the contact map into a 1D signal along
the diagonal for subsequent segmentation or into a graph and apply community
detection algorithms. To run the existing tools, multiplex data must first be
converted into a contact map. However, enumerating over all possible pairs in a
complex is computationally expensive and may introduce additional bias since the
number of pairwise interactions increases quadratic in *n*. In other words, a complex with 5 fragments yields 28 pairs
instead of 1 pair for a complex with 2 fragments. This approach would also lose
valuable multiplexity information.

Conventional studies focused on interactions within these TADs identified computationally. However, a recent Hi-C study has suggested that multiple TADs can interact with each other to accommodate molecular functions during the development (Paulsen et al. [22]). The authors inferred confident domain-wise interactions by finding cliques in a graph, where nodes represent TADs and edges are contact frequency between TADs. Unlike Hi-C datasets, the multiplex data naturally provide interactions among any number of TADs. Thus, it is desirable to exploit this information and assess the statistical significance of these observed inter-TAD interactions.

In parallel, algorithms have been developed to analyze protein-enriched 3D architecture data from assays such as ChIA-PET. Similar to Hi-C, ChIA-PET data are also prone to bias and noise, which are computationally filtered out by statistical algorithms such as ChIA-PET tool (Li et al. [16]) and chiasig (Paulsen et al. [23]). The main idea is to model interaction frequency between two loci as hypergeometric distribution or the non-central hypergeometric distribution. To accommodate recently developed variants HiChIP (Mumbach et al. [21]) and PLAC-seq (Fang et al. [8]), researchers developed hichipper (Lareau and Aryee [15]), fithichip (Bhattacharyya et al. [4]), and MAPS (Juric et al. [11]) to remove systematic biases and identify significant loops. In ChIA-PIPE (Capurso et al. [5]), the de-noising is done by filtering out loops without peak supports in the anchors. Unfortunately, these tools are specifically designed to model interactions between two loci and would not readily generalize to those involving more than two loci.

Thus, to fill in the gap in novel software for analyzing multiplex data,
we developed MIA-Sig (*M*ultiplex *I*nteractions *A*nalysis by *Sig*nal processing
algorithms) with a set of Python modules tailored for ChIA-Drop and related data
types. MIA-Sig has the following components: (1) calling statistically significant
complexes and removing experimental noise, (2) calling TADs on multiplex data, and
(3) identifying meaningful multi-way inter-TAD contacts.

## Results

### Distance test resolves multiplets and removes experimental noise

A central challenge in ChIA-Drop data analysis is to distinguish the true biological chromatin complexes from the experimental noise. One possible source of noise is an event that two or more chromatin complexes are potentially encapsulated in the same microfluidic droplet and then are assigned the same barcode, yielding a multiplet (Fig. 1a). The problem also prevails in microfluidic-based single-cell RNA-seq data, which is then resolved computationally via dimensionality reduction and clustering (Wolock et al. [30]). However, methods developed for single-cell transcriptomics data are not apt for multiplex chromatin interactions data since (1) the signal for chromatin interactions is point data (fragment is captured or not captured) rather than continuously valued data (gene expression level), and (2) multiplex chromatin interaction data are inherently more sparse than the single-cell transcriptomics data, due to the lack of cell barcodes.

Therefore, we devised a distance test with an entropy filter based on the biological knowledge that most meaningful chromatin interactions occur in a certain distance range, while those outside the range are likely noise (Lajoie et al. [14]). By converting the distances between fragments into a probability vector, we compute the normalized Shannon entropy (Shannon [27]), ranging from 0 to 1. If a droplet contains a single complex, the fragments are presumably close and equally spaced, resulting in high entropy close to 1. In the case of a doublet, two independent complexes would be separated by a single large distance, resulting in low entropy close to 0, which can then be separated into two singlets (Fig. 1b). The cutoff threshold is determined by the average normalized Shannon entropy of the expected null distribution as described below.

To identify significant chromatin complexes, a resampling-based
distance test is applied before and after the entropy filter (Fig. 1c; Additional file 1: Figure S1a; the “Methods” section). We verify that the distance distribution
of expected complexes from resampling (computational null) and that of pure DNA
complexes (experimental null) are comparable, with the majority greater than
1 Mbps (Additional file 1: Figure
S1b). Finally, we retained 55,995 statistically significant complexes in the*Drosophila* S2 ChIA-Drop data out of
3,075,926 putative complexes (Additional file 1: Figure S1c). Filtering to retain significant complexes
preserves the TADs along the diagonal of the 2D heat maps, while reducing the
off-diagonal noise (Fig. 1d;
visualization through Juicebox (Durand et al. [7])). A shift in distance distributions from large (original)
to small (significant) supports that meaningful interactions are captured within
10 kb and 1 Mb, mostly from complexes with 5 or more fragments
(Fig. 1e;
Additional file 1: Figure
S2).

Of the significant chromatin complexes, 15,055 (27%) were from the
entropy filtering step that resolved doublets and triplets
(Additional file 1: Figure S3a,b).
For example, of complexes with 3 fragments (in *F*_{3}), 499,613 are identified as “singlets”
due to high entropy, and 284,540 are considered to be “doublets” due to low
entropy. A general trend is that entropy is highest for those without any
splits, lowest for a doublet with a singleton, and increases as the size of
sub-complexes balance to be roughly equal.

Several parameters are fixed or to be chosen in the distance test. As mentioned earlier, the cutoff threshold in the entropy filter is computed for each fragment class based on the null distribution; for reference, some of the values used in this study are summarized in Additional file 1: Figure S3c. In general, the threshold is higher for the class with a high number of fragments than for that with a low number of fragments. Other parameters are to be chosen by the users: false discovery rate (FDR), ratio threshold (ratiothresh) for separating the second largest distance in the entropy test, and the sample size for constructing the null. We benchmarked a few values for some of these parameters and evaluated their effects by recording the number of significant complexes and by performing the two-sided K-S test on fragment-to-fragment distances of the original and significant complexes. As expected, the setting with a lower number of significant complexes had higher K-S statistics, likely because MIA-Sig kept a small portion of the highly confident complexes. Given the same FDR, a ratiothresh of 5 yields more complexes in the significant category and a slightly higher K-S statistics than a ratiothresh of 2. The current default parameters are FDR = 0.1 and ratiothresh = 2, but a more systematic evaluation of “real complexes” will be desirable in the future as more multiplex datasets become available.

### Wavelet-based segmentation method identifies TADs overlapping inactive regions

From the significant complexes, it is desirable to automatically
call TADs for downstream analyses. Many TAD calling algorithms exist for Hi-C
data (Zufferey et al. [32]), yet
all are based on pairwise contacts. To retain multiplexity information, we
developed an algorithm to call TADs directly from the ChIA-Drop data (the
“Methods” section). The idea is to
convert complexes into 1D signal track then apply wavelet transformation (Mallat
[19]) to smooth the signal
while retaining clear change points (Additional file 1: Figure S4a). This approach allows us to identify clear
gaps between TADs, rather than segmenting the genome into consecutive TAD
regions (Additional file 1: Figure
S4b). MIA-Sig called 335 TADs with a wider range of sizes than 513 TADs called
by pairwise “insulation score” (InS) approach; similarly, the gap sizes spanned
a wider range for MIA-Sig TADs than for InS TADs (Additional file 1: Figure S5). Compared to InS TADs, the
MIA-Sig TADs are less likely to overlap active regions characterized by high
H3K27ac and low H3K27me3 (Fig. 1f),
which are known to be the gaps between TADs in *Drosophila* (Rowley et al. [26]). This pattern is observed genome-wide: MIA-Sig TADs have
a higher inactive mark (H3K27me3) than InS TADs, and MIA-Sig gaps have a higher
active mark (H3K27ac) than InS gaps (Additional file 1: Figure S6).

### Binomial test detects frequent interactions among two or more TADs

Most interactions occur within a single TAD, but 23% of significant
complexes also cross two or more TADs (Additional file 1: Figure S7a), consistent with previous
findings (Paulsen et al. [22]).
Thus, we identified frequent interactions involving multiple TADs by counting
the occurrences and performing a binomial test (Additional file 1: Figure S7b; the “Methods” section). A set of TADs with frequent contacts are
ultimately assigned low *p* values
(Additional file 1: Figure S7),
which can guide the researchers to perform validation experiments.

### Enrichment test retains strong interactions involving promoters

Similar to ChIA-PET, ChIA-Drop can also enrich chromatin complexes involving a specific protein, such as RNAPII or CTCF. We implemented an enrichment test to estimate the significance of binding intensity of observed chromatin complexes and retain those with high binding intensity (Fig. 2a; the “Methods” section). An empirical null distribution is generated by placing the observed complex on a random location in the chromosome and recording the binding intensity. We verified that the empirical null and observed distributions differ significantly, with observed shifted to the right of the null (Additional file 1: Figure S8c,d). After the enrichment test, we retain 190,226 significant complexes out of 769,803 complexes (Additional file 1: Figure S8).

These significant complexes have their fragments in highly enriched domains characterized by high RNA-seq expression and H3K27ac signal with abundant RNAPII ChIA-PET loops (Fig. 2b). Genome-wide patterns confirm that significant complexes are biased towards active regions, whereas insignificant complexes are not (Additional file 1: Figure S9). Moreover, significant complexes have higher median H3K27ac signals and lower median H3K27me3 signals than insignificant complexes (Fig. 2c, d). A detailed view around a few genes shows that significant complexes are more likely to retain promoter-centric interactions than insignificant complexes (Fig. 2d; visualization through ChIA-View (Tian et al. [29])). This pattern is prevalent genome-wide, with 69% of significant complexes containing at least one promoter compared to only 30% of insignificant complexes (Fig. 2f). Notably, significant complexes are most likely to capture one active promoter and one or more non-promoters—possibly enhancers—while insignificant complexes are prone to detect interactions among non-promoters (Additional file 1: Figure S10). Among the promoter-involving fragments, those in significant complexes have higher median gene expression than those in insignificant ones.

### Insignificant RNAPII ChIA-Drop complexes emulate non-enriched ChIA-Drop data

As with many experimental protocols, the chromatin immunoprecipitation step is not 100% efficient and typically yields a 20–40% efficiency rate (Tang et al. [28]). Thus, we take advantage of the fact that enriched ChIA-Drop datasets also contain some background signal for chromatin complexes that did not specifically involve the protein of interest, similar to non-enriched ChIA-Drop data. Through the MIA-Sig enrichment test on RNAPII ChIA-Drop data, we can extract the non-enriched complexes from the insignificant complexes, which approximately emulate the ChIA-Drop data (Fig. 2g).

### Distance test can be applied to SPRITE data

We have developed MIA-Sig on ChIA-Drop and RNAPII ChIA-Drop data, but it could also be applied for de-noising multiplex chromatin interactions from other methods, such as SPRITE and GAM.

SPRITE uses three to five rounds of split-and-pool approach to barcode each chromatin complex by combinatorial indexing, with a theoretical assumption that many rounds of splitting and pooling should result in one unique barcode combination per chromatin complex. However, in practice, the split-and-pool process is limited to four to five rounds with a limited set of distinct barcodes, and in each round, potentially hundreds of thousands of chromatin complexes are assigned the same DNA oligo barcode. As a result, there is a certain non-zero probability of multiple complexes receiving an identical barcode combination. These unrelated complexes would be considered technical noise of SPRITE technique, which is somewhat similar to that of ChIA-Drop of unrelated complexes partitioned in the same microfluidic droplet.

As a proof-of-concept, we demonstrate the utility of MIA-Sig by
performing the distance test on SPRITE data (Quinodoz et al. [25]) generated from F121 mouse embryonic
stem cells (GSE114242). The data are pre-processed to convert reads into
fragments of certain sizes and distances, and we selected intra-chromosomal
complexes in chr18 (the “Methods”
section). From the original 487,679 complexes, 11,984 complexes are identified
as significant by the 2 distance tests preceding and following the entropy
filter (Fig. 3a). The 2D contact maps of
original complexes exhibit off-diagonal noise, whereas that of the significant
complexes have the majority of the signal along the diagonal (Fig. 3b). We plot the empirical cumulative
distribution of the fragment-to-fragment distances of original and significant
complexes and observe that significant complexes have shorter distances than
original complexes (Fig. 3c; two-sided
Kolmogorov-Smirnov test statistic = 0.18, *p*
value <2.2 × 10^{−16}). These results indicate that
MIA-Sig can indeed assess the statistical significance of complexes captured by
SPRITE.

## Discussion

Many tools exist for analyzing traditional proximity ligation-based chromatin interaction data, such as Hi-C and ChIA-PET. By contrast, there is a lack of tools to comprehend the data generated by the recently developed multiplex interaction mapping techniques. To fill in this gap, we have developed MIA-Sig that is specifically designed to analyze multiplex chromatin interaction data.

The most significant functionality of MIA-Sig is to de-noise and
identify statistically confident multiplex chromatin complexes in both non-enriched
data and protein-enriched data. We applied an entropy concept from information
theory to identify multiplets in ChIA-Drop and SPRITE data and implemented a simple
yet relatively efficient method to evaluate the enrichment score of each complex in
RNAPII ChIA-Drop data. In addition, we proposed a wavelet-based algorithm to call
TADs on multiplex data. A unique feature of this approach is the ability to clearly
distinguish TADs from gaps, which is of biological relevance in *Drosophila* samples. In particular, it is shown that
TADs and gaps interleave in *Drosophila*, unlike in
human or mouse where gaps are not as critical as they are in *Drosophila* (Rowley et al. [26]). These TADs merely serve as a unit in the downstream
analysis, where we investigate the occurrence of simultaneous interactions among two
or more TADs through the binomial test. A recent study support that these
occurrences are important during development (Paulsen et al. [22]). New algorithmic ideas in this work are
implemented in a publicly available package, along with scripts to generate data QC
plots. Hence, MIA-Sig serves as a comprehensive pipeline including both data quality
control and data analysis.

Although potentially a useful package, MIA-Sig nonetheless has its own drawbacks. One key assumption in the distance test is that a fragment far from the other fragments is likely a droplet contamination resulting in a doublet, a behavior yet to be confirmed experimentally and statistically. As with other TAD calling algorithms for Hi-C data, MIA-Sig’s TAD caller requires a set of parameters such as wavelet level and window size. We provide recommended parameters (Lajoie et al. [14]) for each representative model organism, but have not thoroughly tested due to lack of datasets. A critical pitfall in the inter-TAD binomial test is that we do not normalize the TAD interaction frequency by distance and size. In other words, we expect the closer and larger TADs to interact more frequently than others. Finally, in performing the enrichment test for RNAPII ChIA-Drop data, we do not use a background distribution model and instead draw an empirical null distribution via random sampling. A disadvantage of this approach is the computational cost, which can be demanding for large human datasets.

In sum, all multiplex chromatin interaction data could have a significant level of noise, and the principle nature of the noises is conceptually similar. The algorithm used in MIA-Sig considers general issues that should be applicable to all multiplex data. Although the current version of MIA-Sig is specifically developed based on the ChIA-Drop data, we demonstrated its capability to assess the significance of multiplex chromatin complexes in SPRITE data. With further modification and improvement, MIA-Sig should be directly applicable to any multiplex chromatin interaction data and also allow us to fully characterize similarity and differences between experimental protocols.

## Conclusions

As we enter the era of single-cell and single-molecule 3D genome mapping, it will be imperative to develop algorithms to analyze data from these novel experimental protocols. We have presented an approach to solve the imminent problem of extracting statistically significant complexes from noisy signals, calling TADs, and identifying frequent inter-TAD contacts (Fig. 4). In addition, we offer a practical strategy to extract non-enriched ChIA-Drop from RNAPII ChIA-Drop.

We envision that MIA-Sig will be broadly applicable to any type of
multiplex chromatin interaction data ranging from ChIA-Drop and SPRITE to GAM, under
the aforementioned assumptions and with modifications. Here, we focused on the*Drosophila* ChIA-Drop and RNAPII ChIA-Drop
data as a proof of concept and demonstrated that MIA-Sig filters and retains only
the highly informative complexes and tested its applicability to the mammalian data
generated by SPRITE. Finally, as a publicly available software package, MIA-Sig
provides a valuable algorithmic framework for multiplex chromatin interaction data
to be utilized by the broader scientific community.

## Methods

### Notation

An input dataset contains a set of chromatin complexes, each with
two or more fragments. Let *OC*_{m} be the set of fragments contained in the *m*th “observed complex” (OC), for *m* ∈ {1, 2, …, *M*}, and *n* = |*OC*_{m}| is the size of the set denoting the number of
fragments in a complex. Each fragment *u* is
subscripted by the complex index and superscripted by the fragment index and
encodes the genomic location of its origin expressed as a triplet of chromosome,
start and end positions. The distance *d*
between fragments \( {u}_m^a \) and \( {u}_m^b \) is start(\( {u}_m^b \)) − end(\( {u}_m^a \)), and neighboring (fragment-to-fragment; F2F) distances are
encoded in a vector

and the total distance is *d*_{tot}(OC_{m}) = ∑ **x**_{F2F}(OC_{m}); the probability vector \( {\mathbf{p}}_{\mathrm{F}2\mathrm{F}}\left({\mathrm{OC}}_m\right)=\frac{{\mathbf{x}}_{\mathrm{F}2\mathrm{F}}\left({\mathrm{OC}}_m\right)}{d_{\mathrm{tot}}\left({\mathrm{OC}}_m\right)} \). For example, if an eighth complex \( {OC}_8=\left\{{u}_8^1,{u}_8^2,{u}_8^3\right\} \) contains three fragments (chr2L, 100, 500), (chr2L, 1000,
1500), and (chr2L, 6000, 6500), then **x**_{F2F}(OC_{8}) = [500, 4500],*d*_{tot}(OC_{8}) = 5000,
and \( {\mathbf{p}}_{\mathrm{F}2\mathrm{F}}\left({\mathrm{OC}}_m\right)=\left[\frac{1}{10},\frac{9}{10}\right] \). Finally, we can partition *M* complexes OC_{1},
OC_{2}, …, OC_{M} into *F*_{j}, where *j* is the
number of fragments in a complex (OC_{8} belongs to*F*_{3} since it has
three fragments).

### Distance test for non-enriched multiplex chromatin interactions data

#### Empirical null distribution and first distance test

Assuming that complexes are independent of chromosome, we
perform the distance test separately for each chromosome. Motivated by the
fact that each fragment class *F*_{j} has distinct distributions in F2F distances,
we construct the expected null background distribution by randomly rewiring
fragments. Specifically, all neighboring distances **x**_{F2F}(OC_{m}) for *m* ∈ {1, 2, …, *M*} are
placed in a bucket *B*. For each observed*F*_{j}, we randomly draw *j* − 1 elements (with replacement) from *B* to create 100,000 “expected complexes” (EC)
\( {\mathrm{EC}}_k^j \) for *k* ∈ {1, 2, …, 100,000} and store them in *F*_{j}′. Note that since we only care about the distance
between fragments, we can assume that every fragment starts at (chr, 1, 500)
and each fragment is of equal length. In practice, we store minimum
information to save compute memory (implementation details below). For each
OC_{m} in*F*_{j}, we compare its total F2F distance to
total F2F distance in \( {F}_j^{\prime } \) and record the proportion of expected complexes that have
shorter distances than the observed complexes as the estimated “raw*p* value.” Formally, for a
OC_{m} ∈ *F*_{j},

where 1_{{∗}} is an indicator function.
Assuming that complexes in each fragment class are independent, we
subsequently separate the raw *p* values by*F*_{j} and adjust them for multiple hypothesis
testing using Benjamini-Hochberg method (Benjamini and Hochberg
[3]) with false discovery
rate (FDR) of 0.1. The complexes with adjusted *p* value ≤0.1 are considered to be statistically significant
and are classified as “pass1” (*F*_{j,
pass1}). Of those insignificant complexes with adjusted*p* value > 0.1, we “fail1”
(*F*_{j, fail1}) those with two fragments
(OC_{m} ∈ *F*_{2} with adjusted *p* value > 0.1) and treat others in a
separate category called “defer” (*F*_{j,
def}). These “deferred” complexes are passed onto the entropy
filter to correct for droplet contamination.

#### Entropy filter

Some complexes in the “deferred” category may be due to the
experimental noise that can be computationally detected. Specifically, this
step aims to computationally correct for the undesired phenomenon of a
droplet containing more than one chromatin complex (referred to as “doublet”
for two, “triplet” for three, and “multiplet” for two or more). In
single-cell RNA-seq (scRNA-seq; single-cell transcriptome) experiments, the
outcome of a doublet would be a vector of real numbers indicating average
expression of the two cells. By contrast, ChIA-Drop data only provide binary
values indicating if a fragment was captured or not, with a variable number
of fragments. Therefore, the effect of two complexes accidentally being
encapsulated in a single droplet would be a large distance in the data. This
assumption is based on the observation from Hi-C and ChIA-PET data analysis
that true interactions occur within certain range of genomic span. Our goal
is to identify complexes with one dominating distance between fragments.
Using the probability vector of the neighboring distance, we quantify the
likelihood of a dominating event. Formally, for an observed complex
OC_{m} with*n* fragments and **p**_{F2F}(OC_{m}) = [*p*_{1}, *p*_{2}, …, *p*_{n − 1}], we compute the normalized Shannon
entropy (Shannon [27])

The normalization factor log_{2}(*n* − 1) ensures that *H*_{norm}(** x**) ∈ [0, 1] for any probability vector

**. Generally,**

*x**H*

_{norm}is small when only one or two of

*p*

_{i}of are large, in which case we presume that a complex is a multiplet and need to separate into singlets. For each observed complexes in the “deferred” category, we compare its normalized Shannon entropy to the average normalized Shannon entropy of the expected complexes in the corresponding class; if the former is smaller, then we separate the observed complex at the longest distance interaction. In other words, for OC

_{m}∈

*F*

_{j, def}, if

then OC_{m} is separated into

OC_{m,
1}= \( \left\{{u}_m^1,{u}_m^2,\dots, {u}_m^S\right\} \) and OC_{m,
2} = \( \left\{{u}_m^{S+1},{u}_m^{S+2},\dots, {u}_m^n\right\} \),

where \( d\left({u}_m^S,{u}_m^{S+1}\right)=\max {\mathbf{x}}_{\mathrm{F}2\mathrm{F}}\left({\mathrm{OC}}_m\right) \). Furthermore, if the second largest distance is at least
\( \frac{1}{\tau } \) of the largest distance, we also separate at the second
longest distance. *τ* is a variable
parameter and we set it to 2 in our analyses; the larger the *τ*, the more likelihood of a “second cut”
(implying a triplet). The resulting sub-complexes are placed in *F*_{j, def, filt} and are now subject to the
second distance test. Note that we did not perform any statistical test in
this step, only performed filtering. Also, the Shannon entropy merely serves
as a quantification measure for a single complex and should not be confused
with the heterogeneity of all complexes in the ChIA-Drop data.

#### Second distance test

We repeat the distance test after correcting for possible
doublets and triplets. For a OC_{m, ∗} ∈ *F*_{j, def,
filt}

We adjust raw *p* values using
Benjamini-Hochberg method with false discovery rate (FDR) of 0.1. The
complexes with adjusted *p* value ≤0.1 are
classified as “pass2” (*F*_{j,
pass2}) and others are “fail2” (*F*_{j,
fail2}). A diagram of the distance test is illustrated in
Additional file 1: Figure
S1a.

#### Implementation, results, and analysis

MIA-Sig takes putative chromatin complexes as the input, which
are results of the ChIA-DropBox (Tian et al. [29]) data processing and visualization pipeline. The
“distance test” python (v3.6) script encompasses all parts using the
following packages: numpy, random, statsmodels, itertools, os, and sys. We
used the parameters --gen dm3 --fdr 0.1 --cef 2 --sz 100,000 to run the
script on GSM3347523 dataset, which used 1.8 GB of memory and 13 min of CPU
time. To save memory, we store minimal information for the null, total
distance for expected complexes, and their mean entropy for each fragment
class. Two runs with the same parameters should yield identical results
because we set seeds in the construction step for the expected complexes. By
saving the first 1000 expected complexes for each class in a chromosome, we
can compare our expected null model to the biological null model, which is
the “pure DNA” described in (Zheng et al. [31]). Plotting the neighboring distances, we observed
that both the computational null and pure DNA are unimodal with peaks
between 1 and 10 Mbps for all classes (Additional file 1: Figure S1b). After confirming that our
expected complexes do emulate long-range noise, we obtained detailed
statistics of each step resulting in 55,995 significant complexes
(Additional file 1: Figure
S1c). Complexes in each of the “original,” “significant” (“pass1” + “pass2”)
and “insignificant” (“fail1” + “fail2”) categories are converted into a
.short format by enumerating over all pairs of fragments in a complex. Three
.short files are then converted into .hic files via juicer (v1.7.5) to be
visualized in juicebox. A 5-Mbps window on chr3L shows that the original
data exhibit both the signal and noise, which are separated by MIA-Sig into
significant and insignificant, respectively (Additional file 1: Figure S2a). The original observed
complexes have a bimodal distribution for high fragment classes, which is a
distinct behavior from the null distribution (Additional file 1: Figure S1b, S2b). The density plot
further supports that significant complexes retained short distances or a
mix of short and long distances. By contrast, insignificant complexes are
only comprised of unimodal long distances (Additional file 1: Figure S2b). Consistent with an
observation that high-fragment complexes contribute to the structure more
than the low-fragment complexes (Zheng et al. [31]), MIA-Sig assessed the majority of
high-fragment complexes as significant (Additional file 1: Figure S2c). We next investigated the
effects of the entropy filter, which was designed to remove doublets and
triplets. Of the 1,452,878 complexes in the deferred category ranging from*n* = 3 to *n* = 8, MIA-Sig identified 60% (869,065) to be singlets, 34%
(498,291) to be doublets, and 6% (85,522) to be triplets, yielding 548,342
singletons (*F*_{1})
and 1,573,871 complexes (*F*_{≥2}) (Additional file 1: Figure S3). For each class, singletons
had the highest normalized Shannon entropy, followed by doublets and
triplets. The entropy filter step allowed MIA-Sig to identify additional
15,055 complexes as significant, which amounts to 27% of the total
significant complexes.

### TAD calling for non-enriched multiplex chromatin interactions data

#### Generating 1D signal track

Existing TAD calling algorithms for pairwise Hi-C data
generally fall into two categories: (1) signal segmentation after conversion
from 2D contact maps into 1D tracks measuring interaction intensities along
the genome and (2) community detection directly on the 2D heatmap by
treating each bin as a node on an undirected graph. We take the first
approach and convert our complexes into 1D signal track. A conventional
pairwise approach would enumerate over all pairs of fragments in a complex
and record their spans. However, multi-fragment complexes may
over-contribute since the number of pairs grows quadratically:
\( \left(\genfrac{}{}{0pt}{}{n}{2}\right)=\frac{n\left(n-1\right)}{2} \), where *n* is the number
of fragments in a complex. Instead, we allow each complex to only contribute
linearly in *n* by recording its span
weighted by *n*. More precisely,
coordinates are (chrom(\( {u}_m^1\Big) \), start(\( {u}_m^1 \)), end(\( {u}_m^n \)), *n*) for an
OC_{m} with*n* fragments. We finally obtain a
“weighted complex span coverage” by accumulating the coordinates over all
given complexes.

#### Smoothing and segmentation

Our next task is to segment the 1D track into regions with a
high signal and annotate them as TADs. In an ideal case, we can achieve this
goal by computing the slope of the signal **s**
and by recording critical points where the slope is 0. However, our signal
has a basepair resolution and thus is not smooth, resulting in too many
false critical points. A common way to smooth the signal is by a moving
average window, but using a large window size would lose the resolution and
yield TADs with fuzzy boundaries. Moreover, due to the inherent nature of
TAD sizes, a window size parameter optimal in one region may not be optimal
in another. We avoid this parameter tuning step by instead applying a
discrete wavelet transformation, which decomposes signal into high-frequency
component and low-frequency component (Mallat [19]). Of note, the low-frequency
component generally retains the smoothed version of the signal without
affecting the shape, which is helpful for us to find accurate TAD boundaries
(Additional file 1: Figure
S4). Using this “smoothed” signal, we compute the slope and fine-tune TAD
coordinates.

#### Implementation, results, and analysis

The “tad calling” python (v3.6) script encompasses all parts
using the following packages: numpy, os, scipy, pywt, itertools, and sys. We
used the parameters --cat PASS --bs 1000 –sp drosophila --r dm3 to run the
script on significant complexes from the distance test of GSM3347523
dataset, which used 84 MB of memory and 1 min of CPU time. Before generating
the 1D signal track, we separate 2 fragments if they are more than 100 Kb
apart, based on the upper range of general TAD sizes by organisms (Dekker
and Heard [6]). Coverage was
generated by BEDtools (Quinlan and Hall [24]) using “genomecov” function, and the coverage is
binned into 1-Kb windows via “makewindows” and “map” commands. Signal
smoothing was done by pywt package using the parameters “bior1.1” for
wavelet function and “3” for the level. MIA-Sig called 335 TADs over the 6
chromosomes, with a median size of 200 Kb (Additional file 1: Figure S5a). For a comparison, we also
tested insulation score as follows: .hic file (of all pairs of fragments)
are converted into contact matrices via Juicer’s “dump” function with a
dense matrix option (-d) in the Juicer tool (v1.7.5); insulation score
script (https://github.com/dekkerlab/cworld-dekker/tree/master/scripts/perl) is executed with 100-Kb insulation square size, 100-Kb delta
window size for 10-Kb resolution contact maps with balanced normalization.
Insulation score (InS) called 513 TADs with a median of 150 Kb and did not
call any TADs larger than 500 Kb (Additional file 1: Figure S5b,c). When we examined the gaps
(defined as the regions between 2 TADs, if any), MIA-Sig also had a wider
size range than InS (Additional file 1: Figure S5d,e). For each TAD called by MIA-Sig and InS,
we compute the total H3K27me3 signal and plot the genome-wide behavior
(Additional file 1: Figure
S6a). Overall, MIA-Sig has a higher inactive signal in TADs than InS. The
gap regions in *Drosophila* are known to be
transcriptionally active and should positively correlate with the H3K27ac
signal. We confirm that MIA-Sig has a slightly higher median active signal
than InS (Additional file 1:
Figure S6b). Note that we did not perform any normalization by region size
because both algorithms segment the genome into either a TAD or a gap, so
the region size should also be a feature. Histone marks provide biological
evidence that MIA-Sig TADs are inactive and gaps are active, but ChIA-Drop
fragment counts provide a direct measure of TAD and gap intensities. Using
the BEDtools command “intersect -c,” we count the number of fragments in
each region. MIA-Sig generally captured more fragments in TADs than InS did
(Additional file 1: Figure
S6c) and less fragments in gaps than InS (Additional file 1: Figure S6d). Finally, we annotate each
fragment in significant and insignificant complexes as “TAD” or “gap” as
called by MIA-Sig. For each complex, we count the number of TADs with at
least 2 fragments within each TAD. Only 5% of the insignificant complexes
had fragments in 1 or 2 TADs, and the rest were not contributing to the TAD
structure (Additional file 1:
Figure S7a), validating the observation from 2D heatmaps. By contrast, only
26% of the significant complexes were not in TADs, a majority (51.3%) in
intra-TAD interactions, and many (23%) connected 2 or more TADs. By
observing that 12,884 complexes involve 2 to 21 TADs, we next sought to
characterize if multiple complexes connect the same set of TADs.

### Inter-TAD binomial test for non-enriched multiplex chromatin interaction data

#### Motivation and intuition

Our goal is to evaluate the statistical significance of these TAD combinations based on the frequency of occurrence measured by the number of complexes therein. The problem is simple for a pair of TADs: we may treat a TAD as a ChIA-PET loop anchor and apply tools based on the hypergeometric test. However, our data are now multi-dimensional. For instance, suppose that there are five TADs and five combinations “A-C,” “B-C,” “B-C-D,” “A-B-E,” and “A-D-E” (Additional file 1: Figure S7b). The pair “B-C” appears four times on its own, but also appears three times as a part of the triple “B-C-D.” Moreover, some parts of a combination may appear elsewhere with the same number of TADs: given “B-C-D” and “A-C-D,” “C-D” appears twice. Therefore, we propose a counting scheme based on the occurrence of “expanded pairs.”

#### Methods

The notations used defined in this section are independent from
those in other sections. We let the *i*th
combination be \( {\mathrm{TC}}_i=\left\{{T}_i^1,{T}_i^2,\dots, {T}_i^N\right\} \), where each \( {T}_i^n\in \left\{{\mathrm{TAD}}_1,{\mathrm{TAD}}_2,\dots, {\mathrm{TAD}}_M\right\} \), *N* = ∣ TC_{i}∣ is the number of TADs involved, and we partition each TC_{i}
into the same class *G*_{j} if |TC_{i}| = *j*. All pairs of TADs in *TC*_{i} are in Pa(TC_{i}) = {{*r*, *s*} : *r* ≠ *s*, for *r*, *s* ∈ TC_{i}} and \( \left|\mathrm{Pa}\left(\mathrm{T}{\mathrm{C}}_i\right)\right|=\frac{n\left(n-1\right)}{2} \). For each TC_{i}, we record the number of pairs in the same
class as

and the number of exact appearance in higher class as

Using these two numbers, we compute the appearance of “pairs” in the same class and higher class

Finally, we perform the binomial test with *x*(TC_{i}) as the number of success, \( k\left(\mathrm{T}{\mathrm{C}}_i\right)=\sum \limits_{z\in {G}_j}x(z) \) as the number of trials, the probability of success
hypothesized as \( p=\frac{1}{\mid {G}_j\mid } \); the alternative hypothesis is that the observed
probability is greater than the expected probability *p*. A detailed example is provided using the same notations
(Additional file 1: Figure
S7b).

#### Implementation, results, and analysis

A python script “inter-TAD binomial test” implements the method
using packages numpy, itertools, scipy, statsmodels, os, and sys. Of 6861
unique combinations involving 2 to 21 TADs, 915 (13%) were identified as
statistically significant. An example illustrates that a pair of TADs with a
strong signal in the heatmap and many complexes in the linear view has lower*p* value than that with a weak signal
(Additional file 1: Figure
S7c). Here, we assumed that the frequency of interactions between TADs is
independent of their distance and sizes, and we also did not distinguish
contacts with 2 fragments from those with 10 fragments. These parameters may
be incorporated in the future version.

### Enrichment test for RNAPII-enriched multiplex chromatin interaction data

#### Motivation

The above sections are designed to analyze non-specific multiplex interaction data analogous to the Hi-C data. With an additional step of chromatin immunoprecipitation, protein-enriched multiplex data reveal protein-specific interactions similar to the population average ChIA-PET loops. In a typical ChIA-PET analysis, loops anchored in strong binding peaks are considered to be more reliable than those with weak or no peaks. Extending this notion to the multiplex data, we developed an enrichment test for RNAPII ChIA-Drop data. Our end goal is to retain complexes with fragments in strong binding peaks. One approach is to call peaks and only keep complexes that overlap the peak regions. However, peak calling algorithms have their own model assumptions that may not hold for ChIA-Drop data. Even with accurate peak regions, assigning statistical significance to each complex is not a trivial problem since the null distribution is unclear. Thus, we take an alternative—inevitably the computationally expensive—approach by sampling the background null distribution for each complex.

#### Statistical test

The idea is to take the observed complex and place it on a
random location of the same chromosome and compare the mean coverage between
the observed and the expected. Through many rounds of re-sampling, we obtain
the *p* value by counting the number of
occurrences in which the expected coverage exceeds the observed coverage
(Additional file 1: Figure
S8a). More precisely, for an observed complex \( {\mathrm{OC}}_m=\left\{{u}_m^1,{u}_m^2,\dots, {u}_m^n\right\} \), we randomly draw an integer \( i\in \left\{1,\dots, \mathrm{length}\left(\mathrm{chrom}\right)-\mathrm{start}\left({u}_m^1\right)\right\} \) and the shift \( \delta =\mathrm{start}\left({u}_m^1\right)-i \). The first expected complex is then \( {\mathrm{EC}}_1^m=\left\{{v}_m^1,{v}_m^2,\dots, {v}_m^n\right\} \), where \( \mathrm{start}\left({v}_m^l\right)=\mathrm{start}\left({u}_m^l\right)-\delta \), and \( \mathrm{end}\left({v}_m^l\right)=\mathrm{end}\left({u}_m^l\right)-\delta \) for all *l* ∈ {1, …, *n*}. Repeating
this process 10,000 times, we obtain \( \mathrm{E}{\mathrm{C}}_k^m \) for *k* ∈ {1, …, 10,000}. We can then compute the raw *p* value of
the *m*th observed complex as:

where \( \mathrm{covg}\left(\mathrm{O}{\mathrm{C}}_m\right)=\frac{1}{n}{\sum}_{l=1}^n\frac{\boldsymbol{fcs}\left(\mathrm{start}\left({u}_m^l\right),\mathrm{end}\left({u}_m^l\right)\right)}{\mathrm{end}\left({u}_m^l\right)-\mathrm{start}\left({u}_m^l\right)} \) and ** fcs**(

*x*,

*y*) is the mean “fragment coverage signal” between coordinates

*x*and

*y*. Raw

*p*values are separated by chromosomes and are adjusted via the Benjamini-Hochberg method with a false discovery rate (FDR) of 0.1. The complexes with adjusted

*p*value ≤0.1 are considered to be statistically significant and are classified as “pass”; others are considered insignificant or “fail.”

#### Implementation, results, and analysis

A python script “enrichment test” utilizes the packages numpy,
random, statsmodels, os, and sys. GSM3347525 RNAPII ChIA-Drop data are
pre-processed to exclude fragments mapped to the repetitive regions in the
genome (dm3.rmsk.bed), and 769,803 complexes remain as “GSM3347525NR.” The
most time-consuming part of the algorithm is to obtain the fragment coverage
at a given location, since we need to search for a start and end indexes in
a bedgraph or a bigwig file. With at least
769,803 × 2 × 10,000 = 1.54 × 10^{10}
operations, we realized that python implementations of exact search would be
intractable. As means to reduce the runtime, we store the bedgraph file into
bins of size 10 bp and store only the fourth column “value.” The solution
then turns into a simple lookup operation, yielding an approximation that is
close to the exact solution. Our code is “parallelized” by chromosome, each
using around 5 h CPU time and 230 MB of memory (Additional file 1: Figure S8b). MIA-Sig identified 190,226
complexes (24.7%) as statistically significant. We ensure that our empirical
null distribution does behave randomly by comparing the enrichment scores of
the observed complexes in chr2L with those of 1000 expected complexes
generated for each observed complex (Additional file 1: Figure S8c). Zooming in further, we note
that the histogram of the observed is shifted to the right of the histogram
of the expected null (Additional file 1: Figure S8d). Using the active and inactive regions
defined in (Zheng et al. [31]),
we count the number of fragments therein for significant and insignificant
complexes (Additional file 1:
Figure S9a). For each active and inactive region, we compute the number of
significant complex fragments and their log10 values are plotted
(Additional file 1: Figure
S9b); K-S test supports that significant complexes are indeed more likely to
be in active regions than in inactive regions. By contrast, insignificant
complexes have no bias towards or against active regions
(Additional file 1: Figure
S9c). We define a gene promoter as ± 1 KB from the transcriptional start
site (TSS) annotated by UCSC genome browser. Note that typically ± 250 bp is
used for *Drosophila*, but we extend it to
accommodate ChIA-Drop protocol-specific features. A gene is active (6466
genes) if the total RNA-seq level is greater than 5 and inactive (8874
genes) otherwise. A fragment is “active promoter” if it overlaps the
promoter of an active gene. In general, significant complexes have higher
proportion of promoter fragments than insignificant complexes
(Additional file 1: Figure
S9d), and the skew is more pronounced for active promoters
(Additional file 1: Figure
S9e). Inactive promoters serve as a control, in which both significant and
insignificant complexes display similar patterns in the number of inactive
promoter fragments (Additional file 1: Figure S9f,g).

### Distance test on mouse F121 SPRITE data

We have performed the distance test on SPRITE data (Quinodoz et al. [25]) generated from F121 mouse embryonic stem cells (GSE114242) mapped to the mm9 reference genome. The pre-processing steps of the SPRITE data are the following: (1) extract complexes in chr18, (2) construct fragments by extending a read mapped position by 1000 bp, (3) exclude read mapped position if it is less than 10,000 bp away from the left-adjacent mapped loci, and (4) only retain complexes with 2 to 500 fragments. These parameters are chosen because Quinodoz et al. treat multiple reads in a bin to be 1 read due to PCR duplicates, where the bin sizes are 10 kb, 20 kb, 25 kb, 40 kb, 50 kb, 200 kb, 250 kb, and 1 Mbps. After converting reads into fragments in our standard file format of 1 complex per line, we ran the distance test with the parameters --gen mm9 --fdr 0.1 --cef 2 --sz 10,000. One modification is that during the first distance test, we “fail” the complexes with more than 100 fragments. The resulting master file is used for generating the 2D contact maps for ALL and PASS categories by enumerating all pairs of fragments in a complex (Fig. 3b). Likewise, the empirical cumulative distribution function is plotted for ALL and PASS categories (Fig. 3c).

## Availability of data and materials

The MIA-Sig software is available under the MIT License at GitHub [12]. A version of the source code used in this manuscript is deposited on Zenodo [13]. ChIA-Drop (GSM6647523) and RNAPII ChIA-Drop (GSM3347525) data were downloaded from the Gene Expression Omnibus (GEO) under SuperSeries accession number GSE109355 [31]. SPRITE “mouse_combined_clusters” data were downloaded from the GEO under SuperSeries accession number GSE114242 [25]. A link to the pure DNA ChIA-Drop data and processed files of relevant data is also available through the MIA-Sig GitHub page [12].

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## Acknowledgements

The authors thank all members of the Ruan and Chuang Lab for helpful discussions.

### Review history

The review history is available as Additional file 2.

### Peer review information

Barbara Cheifet was the primary editor of this article and managed its editorial process and peer review in collaboration with the rest of the editorial team.

## Funding

This study is supported by a Jackson Laboratory Director’s Innovation Fund (DIF19000-18-02). YR is funded by 4DN (U54 DK107967) and ENCODE (UM1 HG009409) consortia. YR is also funded by Human Frontier Science Program (RGP0039/2017) and supported by Florine Roux Endowment.

## Author information

### Authors and Affiliations

### Contributions

MK, MZ, and YR conceived the study. MK devised the algorithms and wrote the MIA-Sig Python software with input from all authors. MZ developed and performed the ChIA-Drop experiments. SZT developed and provided the ChIA-View software. SZT and BL contributed parts of the analyses. MK, JHC, and YR wrote the manuscript. All authors read and approved the final manuscript.

### Corresponding author

## Ethics declarations

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### Consent for publication

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### Competing interests

The authors declare that they have no competing interests.

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## Supplementary information

**Additional file 1: Figure
S1.**

Overview of distance test, comparison of
computational and experimental null distribution, and
summary statistics. **Figure
S2.** Characteristics of original, significant,
and insignificant complexes. **Figure
S3.** Effects of the entropy filter. **Figure S4.** Ideas behind the
MIA-Sig TAD calling algorithm. **Figure
S5.** Statistics of TAD and gap sizes called by
MIA-Sig and Insulation Score. **Figure
S6.** Comparison of TADs and gaps by MIA-Sig
and Insulation Score. **Figure
S7.** Inter-TAD binomial test. **Figure S8.** Overview of enrichment
test for RNAPII ChIA-Drop data. **Figure
S9.** Comparison of significant and
insignificant RNAPII complexes. **Figure
S10.** Annotation of fragment as an active
promoter, inactive promoter, or non-promoter. (DOCX 3001 KB)

**Additional file 2.**

Review history.

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## About this article

### Cite this article

Kim, M., Zheng, M., Tian, S.Z. *et al.* MIA-Sig: multiplex chromatin interaction analysis by signal
processing and statistical algorithms.
*Genome Biol* **20**, 251 (2019). https://doi.org/10.1186/s13059-019-1868-z

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DOI: https://doi.org/10.1186/s13059-019-1868-z

### Keywords

- 3D genomics
- Multiplex chromatin interactions
- ChIA-Drop
- Signal processing
- Algorithms