# Dense neural networks for predicting chromatin conformation

**Part of the following topical collections:**

## Abstract

### Background

DNA inside eukaryotic cells wraps around histones to form the 11nm chromatin fiber that can further fold into higher-order DNA loops, which may depend on the binding of architectural factors. Predicting how the DNA will fold given a distribution of bound factors, here viewed as a type of sequence, is currently an unsolved problem and several heterogeneous polymer models have shown that many features of the measured structure can be reproduced from simulations. However a model that determines the optimal connection between sequence and structure and that can rapidly assess the effects of varying either one is still lacking.

### Results

Here we train a dense neural network to solve for the local folding of chromatin, connecting structure, represented as a contact map, to a sequence of bound chromatin factors. The network includes a convolutional filter that compresses the large number of bound chromatin factors into a single 1D sequence representation that is optimized for predicting structure. We also train a network to solve the inverse problem, namely given only structural information in the form of a contact map, predict the likely sequence of chromatin states that generated it.

### Conclusions

By carrying out sensitivity analysis on both networks, we are able to highlight the importance of chromatin contexts and neighborhoods for regulating long-range contacts, along with critical alterations that affect contact formation. Our analysis shows that the networks have learned physical insights that are informative and intuitive about this complex polymer problem.

## Keywords

Chromatin folding Dense neural network HI-C ChIP## Abbreviations

- CNN
Convolutional neural network

- DNN
Dense neural network

## Background

In eukaryotic cells, the condensation of the DNA into chromatin fibers that fold into specific 3D structures brings distant sites of the genome into spatial proximity. These conformations can modulate the expression of genetic information by altering the frequency of interaction between a distant regulatory element such as an enhancer, and the corresponding target gene promoter. The recent advent of high-throughput sequencing technology has allowed the genome-wide measurement of both chromatin structure via Hi-C contact maps [1, 2] as well as the bound locations of a great number of chromatin-associated factors through ChIP-seq methods [3, 4] and additional methodologies [5].

A large body of evidence supports the hypothesis that the spatial arrangement of bound chromatin factors along the DNA strongly influences the probability of chromatin contacts between distant genomic regions [6]. In particular, megabase-sized genomic compartments with similar chromatin states tend to interact with each other [1, 7]. At a finer sub-megabase scale, topologically associated domains [8, 9, 10] flanked by chromatin insulator or boundary elements [10, 11, 12] work as independent genomic units characterized by their self-interaction and repulsion with other genomic regions [13, 14]. Consequently, models that aim to predict how bound chromatin factors influence the folding of chromosomes are now being developed.

Most of the progress towards predicting chromatin conformation from the states of bound factors has come from simulating heterogeneous beads-on-a-string polymers whose bead types correspond to different chromatin states [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. These simulations have been successful in corroborating that interactions between factors together with topological constraints may be responsible for driving chromatin conformation. Nevertheless, calculating the probability of contact between two genomic sites relies on sampling a vast number of polymer configurations. Consequently, exploring the conformational effects of altering the sequence of bound chromatin states is computationally challenging. An alternative Bayesian approach [29], has been successful in predicting the local contact maps from chromatin states without the need for polymer simulations and can rapidly calculate how contact probabilities change when chromatin states are altered. However, the chromatin states that were used as inputs were based on an unsupervised clustering of bound chromatin factors that did not take into account any structural information [30]. It is therefore unlikely that this classification constitutes the best 1D description of the sequence that determines chromatin structure and one may expect to achieve better predictive power by generating a conformation-specific annotation of the sequence of chromatin states. Methods that integrate both chromatin structure and sequence into a unified framework that can rapidly predict the respective contributions of changing sequence or structure are therefore needed.

Multi-layer neural networks that have been around for decades [31] provide the promise of a framework for learning the connections between the sequence of bound factors and chromatin structure. These networks consist of a series of units known as neurons that take in an input signal and have their weighted connections trained to reduce a defined cost function of the output. Many of the networks that are in use today for feature recognition are of a feed-forward structure consisting of a hierarchy of layers. These are dense neural networks (DNN) when they feature a high number of neurons in each layer and are considered to be “deep” when they feature a high number of layers. DNNs can also be coupled to trainable convolutional filters that help to discover important predictive features in the input that often reduce its dimensionality. Such networks are called convolutional neural networks (CNNs). The universal approximation theorem states that under mild assumptions, a feed-forward neural network with a single layer and a finite number of neurons can approximate any continuous function [32]. This capability is one of the main reasons of their great success at modeling complex problems with minimal design input by humans. Their use in bioinformatics has included predicting gene expression [33], the effect of sequence variants [34], predicting secondary structure [35], motif affinity [36] and filling in missing values in the human methylome [37]. In the context of Hi-C data analysis, multi-layer neural networks have been used to generate statistical confidence estimates for chromatin contacts [38] and to enhance the resolution of contact maps [39].

One challenge in using neural networks to model complex phenomenon is that they hide their inner workings behind large combinations of neuronal connections that are often hard to interpret. For this reason, DNNs tend to be seen as black boxes that can perform a great variety of tasks but offer little mechanistic explanation of how the inputs of the model are being used to generate the output. Nevertheless, in recent years a great amount of effort in the DNN community has been directed to developing techniques to infer how information is processed in these models ranging from sensitivity analysis to interpretability [40, 41, 42, 43, 44, 45, 46].

Here we apply dense neural networks (DNNs) to the problem of chromatin conformation. We show that using DNNs one is not only able to predict chromatin conformation from a sequence of DNA-bound chromatin associated factors, but also predict sequence from chromatin conformation. In addition, the model generates a biologically relevant 1D sequence annotation for chromatin states that is optimized to explain chromatin conformation. Furthermore, using sensitivity analysis we explore how the model relates sequence and conformation and unveil key regulatory features behind their connection. Such an approach highlights the importance of sequence neighborhood in structuring chromatin.

## Results and discussion

### Model predictions

*Drosophila Melanogaster*embryos at a resolution of 10 kbp for structure and we use the genome-wide distribution of 50 different bound chromatin-associated factors as sequence (see Methods).) Our aim is to train the neural networks to make predictions at the intra-chromosomal scale, i.e. using sequence data for a region of a chromosome predict the corresponding sub-region of the Hi-C contact map (or vice versa). To predict structure from sequence (what we will call the forward model), we couple a convolutional filter to a DNN (yielding a CNN) where the filter is used to compress the multitude of bound factors into a single 1D sequence that is then fed into the DNN to predict structure (see Fig. 1). This network architecture provides interpretability by giving a learned 1D sequence representation, {

*σ*}, of the highly complex set of bound factors that best predicts structure, which the network calculates as the probability of contact,

*P*(

*c*

_{ij}) between all sites,

*i*and

*j*(see Methods for details). For predicting sequence from structure we use sub-regions of the Hi-C matrix and train the network to predict the corresponding region of the 1D sequence representation (see Methods). We call this the backward model (see Fig. 3a) and it requires a trained forward model to provide a 1D sequence representation that it can learn in going from structure to sequence. Once trained, the backward model can then predict a likely sequence representation from structure alone. We now provide details of how these models were trained and the biologically meaningful results that they generate.

By training the forward model, the convolutional filter has been fit to optimize the predictive power of the high dimensional input sequence of bound factors to the contact map outputs. The learned weights in the convolutional filter, \(W^{0}_{j}\) in Eq. 2, represent the strength of each chromatin factor in determining structure. Figure 2b shows the sorted distribution of these weights for all chromatin factors. The resulting distribution contains a significant amount of biologically relevant information. We find that one set of factors (negative weights) are associated with inactive/heterochromatic factors (H1, H3K27me3, H3K9me1), whereas the other set of factors (positive weights) corresponds to active/euchromatic factors associated with gene bodies (H3K36me2 and H3K36me3), promoters (RNA Polymerase II (RNAPII)), chromatin remodeler (NURF), poised enhancers (H3K4me1) or insulator proteins CP190, Chromator (Chro) and Beaf32 or CTCF, that are also associated with active genes [48, 49]. Interestingly, applying the filter to the input sequences and histogramming the resulting 1D chromatin annotation we find that it has a bimodal nature (Fig. 2c) where the inactive mode is bell-shaped around a value of ∼ 0.2, whereas the active mode is peaked at one. Moreover, histogramming the 1D chromatin state values of sites grouped by gene transcription level, we corroborate that small values (∼ 0.2) correspond to inactive chromatin whereas large values (∼ 1) correspond to active chromatin. This result is suggestive that experiments that ectopically activate or repress a gene could be used to test the structural changes predicted by our model due to “flipping” the state of the corresponding site from inactive to active, or vice-versa. Similarly, grouping sites based on the “chromatin colors” classification [30] we also find that small annotation values correspond to heterochromatic classes while large values correspond to euchromatic ones (see Additional file 1: Figure S1). Thus the fitted filter has naturally grouped the multitude of different chromatin factors into two groupings, with the heterochromatic factors playing a more dominant singular role in shaping structure and the euchromatic factors having a more heterogeneous influence.

We solved the inverse problem (structure to sequence) by fitting an independent multi-layer DNN (see Methods). This model, named the backward model, predicts the 1D chromatin sequence annotation from the forward model using just the local Hi-C contact map as input. The backward model thus predicts a likely sequence of chromatin states that could form that map. Applying the trained backward model to the test set, the individual sequences predicted from the local contact maps had a correlation of 0.66 with the original 1D sequence and a correlation of 0.73 after averaging overlapping windows (Fig. 2d). The predicted profiles visually resemble a smoothened version of the original 1D sequence from the convolutional filter. This was corroborated by performing a Gaussian smoothing on the original 1D sequence, with the resulting smoothed sequence now having a correlation of 0.93 with the backward model prediction (Fig. 2d). Based on this finding, we hypothesize that the forward-model may in fact be doing smoothing internally in the DNN by predicting chromatin contacts based on a local average of the sequence. We tested this by feeding the Gaussian-smoothened 1D sequence from the convolutional filter as input into the DNN layers of the forward model. The predicted contacts from the smoothed sequence showed a correlation of 0.98 with the previously predicted contacts derived from the non-smoothed sequence (shown in Fig. 3b), indicating that the forward-model network generates a similar contact map output from a smoother description of chromatin factors.

### Spatial analysis of conformational effects

In this section we focus on determining what characteristics of the chromatin sequence influence contact maps the most, and vice-versa, what elements of a local contact map are important for inferring sequence. The following analysis thus serves a double purpose: On the one hand, it answers specific questions about chromatin folding. For instance, it quantifies how a particular structure may be altered by making a given genomic site more active or inactive. Secondly, it verifies that the predictions from our neural networks come from a correct representation of the underlying biological mechanisms rather than the exploitation of data artifacts.

*P*(

*c*

_{ij}) is to sequence

*σ*

_{k}by calculating the gradient of the probability of contact of a pair of sites

*∂*

*P*(

*c*

_{ij})/

*∂*

*σ*

_{k}at each site

*k*of the 3

*w*-long input sequence. This quantity can be obtained by the method of back-propagation (see Methods) and highlights how the probability of contact would change upon increasing the value of

*σ*

_{k}(i.e. making the chromatin more active at that location

*k*). Alternatively, gradient values can also be interpreted as how chromatin states must be altered in order to increase the probability of contact. A negative value of the gradient would imply that to increase the contact probability one would have to decrease the value of

*σ*

_{k}, making the state more inactive. Therefore, such analysis highlights the conformational effect that would be expected upon mutating/altering the bound sequence at each particular location of the genome (Fig. 4a).

To examine the general effects of how varying the sequence affects the probability of contact we averaged *∂**P*(*c*_{ij})/*∂**σ*_{k} over the test data. Specifically, for each pair of sites *i* and *j* in the genome separated by a distance *d*=|*j*−*i*|, we calculated *∂**P*(*c*_{ij})/*∂**σ*_{k} for the data in which (*i*,*j*) appear centred in the sequence *w* (for an example, see the pairs of sites (1) and (2) in Fig. 4a). Then we averaged all pairs of sites in the test data that meet these criteria, obtaining a sequence of average gradients for centred sites situated at a distance *d*. In Fig. 4b we show the 〈*∂**P*(*c*_{ij})/*∂**σ*_{k}〉_{d} at each distance *d* as a function of relative position *k*. We find that making the regions between contacting sites more inactive tends to favor more contact (〈*∂**P*(*c*_{ij})/*∂**σ*_{k}〉_{d}<0). This is strongest at shorter distances of contact (*d*<300 kbp) and becomes weaker as the distance increases. On the other hand, making the sites of contact more active also increases the contact probability (〈*∂**P*(*c*_{ij})/*∂**σ*_{k}〉_{d}>0), regardless of distance of contact. The heat map also shows that active chromatin immediately outside *i* and *j* at shorter ranges of contact can also increase the probability of contact.

The gradient squared (*∂**P*(*c*_{ij})/*∂**σ*_{k})^{2} highlights where the magnitude of the gradient is the greatest, and indicates which sequence locations dominate contact probabilities. In the inset of Fig. 4b we observe that, overall, the chromatin states at the sites of contact are the strongest determinants of contact probability. Nevertheless, the contact probability of sites situated at a shorter distance (*d*<300 kbp) are also strongly determined by the state of the chromatin neighbours in between and outside the sites.

*∂*

*σ*

_{k}/

*∂*

*P*(

*c*

_{ij}), and indicates how a change in contact between two sites

*i*and

*j*would be reflective of a change in the chromatin state at site

*k*. We evaluated

*∂*

*σ*

_{k}/

*∂*

*P*(

*c*

_{ij}) at each genomic location and it can be visualized as a map of gradients with the same size as the contact map. In Fig. 5a we show at the top the contact map and sequence of a particular genomic location. At the bottom of Fig. 5a, we evaluate the gradient

*∂*

*σ*

_{k}/

*∂*

*P*(

*c*

_{ij}) for different

*σ*

_{k}positions in the same genomic location. The heat maps indicate how the chromatin state

*σ*

_{k}would change when increasing a particular

*P*(

*c*

_{ij}). In Fig. 5b, we averaged over the test set the gradients for those

*w*-long sequences whose central chromatin state was either inactive or active. This calculation highlights that when the chromatin state is inactive, an increase in contact between sites situated at the left and the right of the inactive state would be indicative of the inactive state becoming even less active (or more inactive). The opposite trend is held for active sites, with an increase of contact between neighbouring sites corresponding to the site being more active. Last, by looking at the average squared gradient we find that the contacts between sites on the left and right of the site of interest are the main determinants of the chromatin state at that site.

## Conclusion

In this paper we have presented a method using dense neural networks (DNN) for predicting chromatin contact maps from sequences bound by chromatin factors and vice versa. Notably, although a certain amount of human-guided design choices went into structuring the DNN, a large part of the model selection behind it was done automatically. By fitting a convolutional filter, we were able to reduce the high dimensional input of multiple chromatin factors to a single 1D chromatin state sequence that was most predictive of structure. Furthermore, by building an inverse model that predicts the chromatin state sequence from contact maps, we could show that the 2D contact maps can effectively be compressed into a 1D sequence and decompressed back. This supports the theory that chromatin folding (for scales at least < 800 kbp) is strongly determined by chromatin states arising from a sequence of bound factors.

By analyzing how varying the inputs to the two neural networks (forward and backward models) changed the outputs, we highlighted that chromatin conformation is a non-local problem; the probabilities of contact between each pair of sites depends on the chromatin states in the larger neighbourhood. In general, the presence of inactive chromatin between two contacting sites or active chromatin on the sites themselves and outside their flanks increases the probability of contact. Nevertheless, the probability of contact between sites situated at a larger distance (> 300 kbp) is largely determined by the chromatin states of the contacting sites themselves.

The work presented in this paper is a proof-of-concept that can easily be extended to capture more biological features of interest. For instance, one could use a larger number of convolutional filters, which would provide a richer biological description of chromatin states (i.e. not just a simple ’inactive’ versus ’active’ state reduction as was done here). This could also include going to finer resolutions both at the level of bound factors and the contact map [12, 50], and there is no reason why they have to be at the same spatial scale. One could also introduce additional types of sequence as inputs, such as genomic annotations, gene expression measurements, genomic mapability and other relevant information to the problem. In addition, one may also be able to introduce non-sequential inputs, that may allow the inclusion of experimental details such as developmental times, cell types or temperatures, and thus allow the modeling of a heterogeneous mixture of cells.

## Methods

### Hi-C data

Chromatin structural information comes in the form of genome-wide contact maps that were obtained from the publicly available Hi-C experiments done by Schuettengruber et al. [51] (GSE61471), performed on 3000-4000 *Drosophila melanogaster* embryos, 16-18 hours after egg laying. The contact map is an array whose elements *n*_{ij} are the number of times a particular pair of genomic sites were found to be in contact. The contact map was built using a set of *N* non-overlapping sites of a fixed size (we used a size of 10 kbp which gave *N*=9663 for all the autosomal chromosomes of *D. melanogaster*). The counts, *n*_{ij}, between all pair of sites *i* and *j* in the contact map were determined by counting up all sequenced pairs from the Hi-C measurement that fell into a given pair yielding an *N*×*N* symmetric matrix (and each unique sequence pair was only counted once to reduce experimental bias, as suggested in [8]).

We normalized the contact map using the ICE method [52] so that total number of counts along each row across the contact map was the same. Then, we measured the average number of contacts at each distance of contact and divided the Hi-C counts by it. This correction removed the strong decaying signal as a function of the distance between contacting sites due to the entropic polymer effect (such as done in [53]). Our final contact maps, which we label as *P*(*c*_{ij}), correspond to contact enrichments at a given distance and are proportional to the actual probabilities of contact when the polymer entropy is removed.

### ChIP-seq data

For the sequence of bound factors, we used the enriched genomic regions of 50 chromatin factors measured with ChIP-seq in 14-16 hour *D. melanogaster* embryos [54]. Specifically, we downloaded the following factors: BEAF, H3K23ac, H3K79Me1, HP1, POF, CP190, H3K27Ac, H3K79me2, HP1b, Pc, CTCF, H3K27me2, H3K79me3, HP1c, Psc, Chro, H3K27me3, H3K9acS10P, HP2, RNA Pol II, GAF, H3K36me1, H3K9me1, HP4, RPD3, H1, H3K36me2, H3K9me2, JHDM1, SU(HW)-HB, H2AV, H3K36me3, H3K9me3, LSD1, Su(var)3, H2B-ubiq, H3K4me1, H4, MBD-R2, ZW5, H3, H3K4me1, H4K16ac(M), MOF, dMi, H3K18Ac, H3K4me3, H4K20me1, NURF301, dRING. This data is publicly available at http://www.modencode.org/ as part of the modENCODE project.

Using the same genomic binning that was used in constructing the contact map (size of 10 kbp), we built *M* (here *M*=50) sequence profiles of length *N* (here *N*=9663) by calculating what fraction of each site was enriched for a given chromatin factor. Therefore, the values of the sequence profiles range from 0 (factor is not present) to 1 (the bin is fully occupied by the factor).

### Transcription data

Gene transcription data was obtained from publicly available RNA tag sequences detected with Illumina GAII with the digital gene expression (DGE) module from duplicate RNA samples from Kc167 cells [30](GSE22069).

We assigned a transcription score to each genomic bin by multiplying RNA counts by the fraction of the genomic bin that is occupied by the the gene in question. Next, we classified each bin into either active of inactive by tresholding transcription scores at one (inactive: transcription score < 1, active: transcription score > 1).

### Dense neural networks for connecting conformation to sequence

A schematic for our model that uses a convolutional neural network (CNN) to predict chromatin contact maps from bound-DNA sequence data is shown in Fig. 1. The output of the network is a local contact map of size *w*×*w* that contains *w*(*w*+1)/2 independent elements (we take *w*=80 that gives 3240 network outputs). With respect to input, based on our prior work [29] that showed the importance of flanking sequence neighborhoods, we take the sub-sequence of length *w* from the *M* sequence profiles that is centered on the *w* sites of the contact map, along with flanking sequences of size *w* giving an input array of size *M*×3*w*.

*M*×3

*w*) input reducing its dimensionality to a one-dimensional 3

*w*-long vector (Fig. 1) whose individual values range from 0 to 1. This vector can be interpreted as a one-dimensional sequence of chromatin states (with values between 0 and 1) that is used as input to the rest of the neural network to predict contact maps. More specifically, if we denote by \(\vec {x}_{i}\) the

*i*th position of the input sequence, with dimension

*M*equal to the number of chromatin factors, the value of the 1D chromatin annotation at that position,

*σ*

_{i}, is obtained from

where \(W^{0}_{j}\) and *β*_{0} are the trainable weights of the convolutional filter. The index *j* corresponds to each of the *M* chromatin factors. The fitted filter thus denotes the weights applied to each of the *M* bound factors for classifying the chromatin into a single sequence that is the best predictor of structure.

*w*is fed to a DNN with multiple layers of increasing size, where the last layer has

*w*×(

*w*+1)/2 outputs corresponding to the values of a local contact map for

*w*bins. The value obtained at neuron

*i*of layer

*n*, \(y^{n}_{i}\), is calculated using the values of all neurons

*k*in the previous layer \(y^{n-1}_{k}\),

where \(W^{n}_{k}\) is a matrix of weights applied to each neuron of the previous layer, *β*_{i} is a constant, and *ReLu* is the rectified linear unit function, namely *f*(*x*)=max(0,*x*), which helps to introduce non-linearities and sparse activation (50*%* of neurons are activated) while remaining easily computed and differentiated [55]. Both \(W^{n}_{k}\) and *β*_{i} are trainable parameters.

The cost function to be minimized during the fitting procedure was taken to be the mean squared error between experimental and predicted distance-normalized contact maps, along with L2 regularization of the filter weights. The technique of dropout regularization, that consists on setting the output of randomly selected neurons to zero with a given probability was used to control for over-fitting [56]. Optimization was done using stochastic gradient descent. We use the Python package Keras ( https://github.com/keras-team/keras ) to code our model, running on top of TensorFlow ( https://www.tensorflow.org/ ).

For the particular example used in the Results the following network was built and fit. The output from the convolutional filter is fed to four ReLu-activated layers of exponentially increasing size, where the last layer is the output layer with same size as the output chromatin map data (Layer 1: 460 neurons, Layer 2: 881 neurons, Layer 3: 1690 neurons, Output Layer: 3240 neurons). A dropout of 0.1 was applied to the dense layers during training. The training was divided into 30 batches, only evaluating the cost function on one batch at a time. At the end of each epoch an independent validation set was used to evaluate the cost function independently to avoid over-fitting. The fitting procedure ended when the cost function calculated in the validation set converged. Results were then calculated on the test data set. The training converged in approximately 30 minutes on a personal laptop.

In addition, we also built a dense neural network (DNN) that solves the inverse problem. Namely, it is trained to predict the previously found 1D chromatin annotation from contact maps alone. The architecture of this network resembles an inverted version of the forward-model, and we thus name it “the backward model”. This network outputs a *w*-long vector of 1D chromatin states from the *w*×(*w*+1)/2 contacts between pairs of sites in the sequence window. Note that the output of the backward model is *w*-long in contrast to the 3*w*-long sequence used as input of the forward model. This is because the flanking regions are hard to predict using just the contact map from the *w*-long interior region, and trying to predict the 3*w* long sequence leads to convergence errors in the procedure. The network is comprised of multiple ReLu-activated dense layers, except for the last output layer which is sigmoid-activated. The “backward model” presented in Results was made of three layers of exponentially decreasing size, the first two ReLu-activated, and the last layer (output layer) is sigmoid-activated (Layer 1: 943 neurons, Layer 2: 274 neurons, Output Layer: 80 neurons). It was fit using the same procedure as the feed-forward network.

### Datasets for training, validating and testing

For training and validating the models, we have used sequence and contact map data from the *D. melanogaster* chromosomes 2L, 2R, 3L and the first half of chromosome 3R (from 1 to 12.95 Mbp). From these regions we obtained 13814 pairs of local sequences and structures using *w*=80. We randomly subsampled 80% of this data as a training set (11052 pairs) and 20% was set aside as a validation set that was not used in the parameter fitting procedure (2768 pairs). For testing the predictions of the model, we used the remaining second half of chromosome 3R (from 14.95 to 26.91 Mbp) as a test set, which contained 2112 pairs of sequences and structures. (It should be noted that our datasets included left-right inverted versions of the data, as the directionality of the genome should not influence the relationship between chromatin contacts and sequence. This thus allowed us to build a dataset of sequences and structures with a size approximately twice the length of the binned genome.)

### Gradient analysis of DNNs

We calculated gradients of the network models using a method known as sensitivity analysis [57, 58]. In particular, we followed the DeepTaylor tutorial in www.heatmapping.org to calculate for a given output neuron the gradient of the output function with respect to the input variables. First, we exported the values of every layer of our trained neural network to text files. Then, for each neuron in the last layer that we wanted to calculate the gradient of, we rebuilt the trained neural network and only included the neuron of interest in the last layer. These neural networks were built using a minimal neural network implementation that can be found in the script “modules.py” from www.heatmapping.org/tutorial . The back-propagation of the gradient of the neuron in the last layer with respect to the input data was done using the methods in “utils.py” from www.heatmapping.org/tutorial as described in the website tutorial.

## Notes

### Acknowledgements

PF would like to thank Cory Simon for many helpful discussions.

### Funding

This work was supported by NSERC to EE’s lab and PF and by FRM grant DEQ20160334940 to OC’s lab and AH. The Funding agencies did not have any role in the design, collection, analysis or interpretation of the data or writing of the manuscript.

### Availability of data and materials

The ChIP-seq datasets analyzed for this paper are available at modencode http://modencode.org . The HiC dataset is available at GEO with reference ID, GSE61471.

### Authors’ contributions

PF, EE conceived the work and PF carried out all analysis. PF and EE wrote the paper. AH, OC helped with data curation, biological interpretation and writing of the paper. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary material

## References

- 1.Lieberman-Aiden E, van Berkum NL, Williams L, Imakaev M, Ragoczy T, Telling A, Amit I, Lajoie BR, Sabo PJ, Dorschner MO, Sandstrom R, Bernstein B, Bender MA, Groudine M, Gnirke A, Stamatoyannopoulos J, Mirny LA, Lander ES, Dekker J. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science. 2009; 326(5950):289–93.CrossRefGoogle Scholar
- 2.Van Berkum NL, Lieberman-Aiden E, Williams L, Imakaev M, Gnirke A, Mirny LA, Dekker J, Lander ES. Hi-C: a method to study the three-dimensional architecture of genomes. J Visualized Exp JoVE. 2010; 39:1869.Google Scholar
- 3.Johnson DS, Mortazavi A, Myers RM, Wold B. Genome-wide mapping of in vivo protein-DNA interactions. Science. 2007; 316(5830):1497–502.CrossRefGoogle Scholar
- 4.Park PJ. ChIP-seq: advantages and challenges of a maturing technology. Nat Rev Genet. 2009; 10(10):669.CrossRefGoogle Scholar
- 5.Cuvier O, Fierz B. Dynamic chromatin technologies: from individual molecules to epigenomic regulation in cells. Nat Rev Genet. 2017; 18(8):457–72.CrossRefGoogle Scholar
- 6.Bonev B, Cavalli G. Organization and function of the 3D genome. Nat Rev Genet. 2016; 17(11):661–78.CrossRefGoogle Scholar
- 7.Rao SS, Huntley MH, Durand NC, Stamenova EK, Bochkov ID, Robinson JT, Sanborn AL, Machol I, Omer AD, Lander ES, et al.A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping. Cell. 2014; 159(7):1665–80.CrossRefGoogle Scholar
- 8.Sexton T, Yaffe E, Kenigsberg E, Bantignies F, Leblanc B, Hoichman M, Parrinello H, Tanay A, Cavalli G. Three-dimensional folding and functional organization principles of the Drosophila genome. Cell. 2012; 148(3):458–72.CrossRefGoogle Scholar
- 9.Nora EP, Lajoie BR, Schulz EG, Giorgetti L, Okamoto I, Servant N, Piolot T, van Berkum NL, Meisig J, Sedat J, et al.Spatial partitioning of the regulatory landscape of the X-inactivation centre. Nature. 2012; 485(7398):381–5.CrossRefGoogle Scholar
- 10.Dixon JR, Selvaraj S, Yue F, Kim A, Li Y, Shen Y, Hu M, Liu JS, Ren B. Topological domains in mammalian genomes identified by analysis of chromatin interactions. Nature. 2012; 485(7398):376–80.CrossRefGoogle Scholar
- 11.Vogelmann J, Valeri A, Guillou E, Cuvier O, Nollmann M. Roles of chromatin insulator proteins in higher-order chromatin organization and transcription regulation. Nucleus. 2011; 2(5):358–69.CrossRefGoogle Scholar
- 12.Stadler MR, Haines JE, Eisen MB. Convergence of topological domain boundaries, insulators, and polytene interbands revealed by high-resolution mapping of chromatin contacts in the early drosophila melanogaster embryo. eLife. 2017; 6:e29550.CrossRefGoogle Scholar
- 13.Dekker J. Two ways to fold the genome during the cell cycle: insights obtained with chromosome conformation capture. Epigenetics Chromatin. 2014; 7(1):25.CrossRefGoogle Scholar
- 14.Dixon JR, Gorkin DU, Ren B. Chromatin domains: the unit of chromosome organization. Mol Cell. 2016; 62(5):668–80.CrossRefGoogle Scholar
- 15.Ganai N, Sengupta S, Menon GI. Chromosome positioning from activity-based segregation. Nucleic Acids Res. 2014; 42(7):4145–59.CrossRefGoogle Scholar
- 16.Bohn M, Heermann DW, van Driel R. Random loop model for long polymers. Phys Rev E. 2007; 76(5):051805.CrossRefGoogle Scholar
- 17.Jost D, Carrivain P, Cavalli G, Vaillant C. Modeling epigenome folding: formation and dynamics of topologically associated chromatin domains. Nucleic Acids Res. 2014; 42(15):9553–61.CrossRefGoogle Scholar
- 18.Sanborn AL, Rao SS, Huang S-C, Durand NC, Huntley MH, Jewett AI, Bochkov ID, Chinnappan D, Cutkosky A, Li J, et al.Chromatin extrusion explains key features of loop and domain formation in wild-type and engineered genomes. Proc Natl Acad Sci. 2015; 112(47):6456–65.CrossRefGoogle Scholar
- 19.Di Pierro M, Zhang B, Aiden EL, Wolynes PG, Onuchic JN. Transferable model for chromosome architecture. Proc Natl Acad Sci. 2016; 113(43):12168–73.CrossRefGoogle Scholar
- 20.Michieletto D, Orlandini E, Marenduzzo D. Polymer model with epigenetic recoloring reveals a pathway for the de novo establishment and 3D organization of chromatin domains. Phys Rev X. 2016; 6(4):041047.Google Scholar
- 21.Mukhopadhyay S, Schedl P, Studitsky VM, Sengupta AM. Theoretical analysis of the role of chromatin interactions in long-range action of enhancers and insulators. Proc Natl Acad Sci. 2011; 108(50):19919–24.CrossRefGoogle Scholar
- 22.Michieletto D, Marenduzzo D, Wani AH. Chromosome-wide simulations uncover folding pathway and 3D organization of interphase chromosomes. 2016; arXiv:1604.03041.Google Scholar
- 23.Barbieri M, Chotalia M, Fraser J, Lavitas L-M, Dostie J, Pombo A, Nicodemi M. Complexity of chromatin folding is captured by the strings and binders switch model. Proc Natl Acad Sci. 2012; 109(40):16173–8.CrossRefGoogle Scholar
- 24.Brackley CA, Taylor S, Papantonis A, Cook PR, Marenduzzo D. Nonspecific bridging-induced attraction drives clustering of DNA-binding proteins and genome organization. Proc Natl Acad Sci. 2013; 110(38):3605–11.CrossRefGoogle Scholar
- 25.Benedetti F, Dorier J, Burnier Y, Stasiak A. Models that include supercoiling of topological domains reproduce several known features of interphase chromosomes. Nucleic Acids Res. 2013; 42(5):2848–55.CrossRefGoogle Scholar
- 26.Brackley C, Johnson J, Kelly S, Cook P, Marenduzzo D. Binding of bivalent transcription factors to active and inactive regions folds human chromosomes into loops, rosettes and domains. 2015; arXiv:1511.01848.Google Scholar
- 27.Brackley CA, Babbs C, Marenduzzo D, Waithe D, Davies J, Brown JM, Hughes JR, Buckle VJ. Predicting the three-dimensional folding of cis-regulatory regions in mammalian genomes using bioinformatic data and polymer models. Genome Biol. 2016; 17(1):59.CrossRefGoogle Scholar
- 28.Di Pierro M, Cheng RR, Aiden EL, Wolynes PG, Onuchic JN. De novo prediction of human chromosome structures: Epigenetic marking patterns encode genome architecture. Proc Natl Acad Sci. 2017; 114:12126–31.CrossRefGoogle Scholar
- 29.Farré P, Emberly E. A maximum-entropy model for predicting chromatin contacts. PLoS Comput Biol. 2018; 14(2):1005956.CrossRefGoogle Scholar
- 30.Filion GJ, van Bemmel JG, Braunschweig U, Talhout W, Kind J, Ward LD, Brugman W, de Castro IJ, Kerkhoven RM, Bussemaker HJ, van Steensel B. Systematic protein location mapping reveals five principal chromatin types in Drosophila cells. Cell. 2010; 143(2):212–24.CrossRefGoogle Scholar
- 31.Schmidhuber J. Deep learning in neural networks: An overview. Neural Netw. 2015; 61:85–117.CrossRefGoogle Scholar
- 32.Hornik K. Approximation capabilities of multilayer feedforward networks. Neural Netw. 1991; 4(2):251–7.CrossRefGoogle Scholar
- 33.Singh R, Lanchantin J, Robins G, Qi Y. Deepchrome: deep-learning for predicting gene expression from histone modifications. Bioinformatics. 2016; 32(17):639–48.CrossRefGoogle Scholar
- 34.Zhou J, Troyanskaya OG. Predicting effects of noncoding variants with deep learning–based sequence model. Nat Methods. 2015; 12(10):931–4.CrossRefGoogle Scholar
- 35.Zhou J, Troyanskaya OG. Deep supervised and convolutional generative stochastic network for protein secondary structure prediction. 2014; arXiv:1403.1347.Google Scholar
- 36.Alipanahi B, Delong A, Weirauch MT, Frey BJ. Predicting the sequence specificities of dna-and rna-binding proteins by deep learning. Nat Biotechnol. 2015; 33(8):831–8.CrossRefGoogle Scholar
- 37.Angermueller C, Lee HJ, Reik W, Stegle O. Deepcpg: accurate prediction of single-cell dna methylation states using deep learning. Genome Biol. 2017; 18(1):67.CrossRefGoogle Scholar
- 38.Schreiber J, Libbrecht M, Bilmes J, Noble W. Nucleotide sequence and DNaseI sensitivity are predictive of 3D chromatin architecture. bioRxiv. 2017:103614. https://doi.org/10.1101/103614.
- 39.Zhang Y, An L, Xu J, Zhang B, Zheng WJ, Hu M, Tang J, Yue F. Enhancing Hi-C data resolution with deep convolutional neural network HiCPlus. Nat Commun. 2018; 9(1):750.CrossRefGoogle Scholar
- 40.Montavon G, Samek W, Müller K-R. Methods for interpreting and understanding deep neural networks. Digit Signal Process. 2018; 73:1–15.CrossRefGoogle Scholar
- 41.Simonyan K, Vedaldi A, Zisserman A. Deep inside convolutional networks: Visualising image classification models and saliency maps. 2013; abs/1312.6034.Google Scholar
- 42.Zeiler MD, Fergus R. Visualizing and Understanding Convolutional Networks In: Fleet D, Pajdla T, Schiele B, Tuytelaars T, editors. Computer Vision – ECCV 2014. ECCV 2014. Lecture Notes in Computer Science. vol 8689. Cham: Springer: 2014.Google Scholar
- 43.Arras L, Horn F, Montavon G, Müller K-R, Samek W. What is relevant in a text document?: An interpretable machine learning approach. PLoS ONE. 2017; 12(8):0181142.CrossRefGoogle Scholar
- 44.Finnegan A, Song JS. Maximum entropy methods for extracting the learned features of deep neural networks. PLoS Comput Biol. 2017; 13(10):1005836.CrossRefGoogle Scholar
- 45.Taylor BJ. Methods and Procedures for the Verification and Validation of Artificial Neural Networks. Boston: Springer; 2006.Google Scholar
- 46.Baehrens D, Schroeter T, Harmeling S, Kawanabe M, Hansen K, MÃžller K-R. How to explain individual classification decisions. J Mach Learn Res. 2010; 11(Jun):1803–31.Google Scholar
- 47.Hothorn T, Lausen B. Double-bagging: Combining classifiers by bootstrap aggregation. Pattern Recognit. 2003; 36(6):1303–9.CrossRefGoogle Scholar
- 48.Emberly E, Blattes R, Schuettengruber B, Hennion M, Jiang N, Hart CM, Käs E, Cuvier O. BEAF regulates cell-cycle genes through the controlled deposition of H3K9 methylation marks into its conserved dual-core binding sites. PLoS Biol. 2008; 6(12):327.CrossRefGoogle Scholar
- 49.Bushey AM, Ramos E, Corces VG. Three subclasses of a Drosophila insulator show distinct and cell type-specific genomic distributions. Genes Dev. 2009; 23(11):1338–50.CrossRefGoogle Scholar
- 50.Eagen KP, Hartl TA, Kornberg RD. Stable chromosome condensation revealed by chromosome conformation capture. Cell. 2015; 163(4):934–46.CrossRefGoogle Scholar
- 51.Schuettengruber B, Elkayam NO, Sexton T, Entrevan M, Stern S, Thomas A, Yaffe E, Parrinello H, Tanay A, Cavalli G. Cooperativity, specificity, and evolutionary stability of Polycomb targeting in Drosophila. Cell Rep. 2014; 9(1):219–33.CrossRefGoogle Scholar
- 52.Imakaev M, Fudenberg G, McCord RP, Naumova N, Goloborodko A, Lajoie BR, Dekker J, Mirny LA. Iterative correction of Hi-C data reveals hallmarks of chromosome organization. Nat Methods. 2012; 9(10):999–1003.CrossRefGoogle Scholar
- 53.Yaffe E, Tanay A. Probabilistic modeling of Hi-C contact maps eliminates systematic biases to characterize global chromosomal architecture. Nat Genet. 2011; 43(11):1059–65.CrossRefGoogle Scholar
- 54.Roy S, Ernst J, Kharchenko PV, Kheradpour P, Negre N, Eaton ML, Landolin JM, Bristow CA, Ma L, Lin MF, et al.Identification of functional elements and regulatory circuits by Drosophila modENCODE. Science. 2010; 330(6012):1787–97.CrossRefGoogle Scholar
- 55.Hahnloser RH, Sarpeshkar R, Mahowald MA, Douglas RJ, Seung HS. Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature. 2000; 405(6789):947–51.CrossRefGoogle Scholar
- 56.Hinton GE, Srivastava N, Krizhevsky A, Sutskever I, Salakhutdinov RR. Improving neural networks by preventing co-adaptation of feature detectors. 2012; arXiv:1207.0580.Google Scholar
- 57.Gevrey M, Dimopoulos I, Lek S. Review and comparison of methods to study the contribution of variables in artificial neural network models. Ecol Model. 2003; 160(3):249–64.CrossRefGoogle Scholar
- 58.Montavon G, Lapuschkin S, Binder A, Samek W, Müller K-R. Explaining nonlinear classification decisions with deep Taylor decomposition. Pattern Recog. 2017; 65:211–22.CrossRefGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.