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normGAM: an R package to remove systematic biases in genome architecture mapping data
BMC Genomics volume 20, Article number: 1006 (2019)
Abstract
Background
The genome architecture mapping (GAM) technique can capture genomewide chromatin interactions. However, besides the known systematic biases in the raw GAM data, we have found a new type of systematic bias. It is necessary to develop and evaluate effective normalization methods to remove all systematic biases in the raw GAM data.
Results
We have detected a new type of systematic bias, the fragment length bias, in the genome architecture mapping (GAM) data, which is significantly different from the bias of window detection frequency previously mentioned in the paper introducing the GAM method but is similar to the bias of distances between restriction sites existing in raw HiC data. We have found that the normalization method (a normalized variant of the linkage disequilibrium) used in the GAM paper is not able to effectively eliminate the new fragment length bias at 1 Mb resolution (slightly better at 30 kb resolution). We have developed an R package named normGAM for eliminating the new fragment length bias together with the other three biases existing in raw GAM data, which are the biases related to window detection frequency, mappability, and GC content. Five normalization methods have been implemented and included in the R package including KnightRuiz 2norm (KR2, newly designed by us), normalized linkage disequilibrium (NLD), vanilla coverage (VC), sequential component normalization (SCN), and iterative correction and eigenvector decomposition (ICE).
Conclusions
Based on our evaluations, the five normalization methods can eliminate the four biases existing in raw GAM data, with VC and KR2 performing better than the others. We have observed that the KR2normalized GAM data have a higher correlation with the KRnormalized HiC data on the same cell samples indicating that the KRrelated methods are better than the others for keeping the consistency between the GAM and HiC experiments. Compared with the raw GAM data, the normalized GAM data are more consistent with the normalized distances from the fluorescence in situ hybridization (FISH) experiments. The source code of normGAM can be freely downloaded from http://dna.cs.miami.edu/normGAM/.
Introduction
The HiC technique uses different restriction enzymes and proximitybased ligation to capture genomewide chromatin interactions [1, 2], which provides largescale maps of the threedimensional (3D) architecture of the whole genome. Similar to the HiC experiments [1], the genome architecture mapping (GAM) experiments [3] can also capture genomewide chromatin proximities. However, GAM has the following advantages compared with HiC [1]: (1) GAM only needs ultrathin cryosectioning instead of ligation; (2) GAM experiments can detect triplet contacts between multiple chromatin regions more effectively than the HiC experiments; (3) GAM only needs hundreds of cells compared with millions of cells needed in population HiC experiments [2,3,4]. Although it is not the focus of this study, it is important to mention that singlecell HiC technique has been invented [5, 6] to capture the DNA proximities of individual cells, based on which 3D genome structures of individule cells can be reconstructed [7]. GAM has been included in the 4D Nucleome (4DN) Network [8], can be visualized using 3D Genome Browser [9], and has been used to assess the accuracy of 3D chromatin reconstructions [10]. All of the interacting capture techniques including HiC, the variants of HiC such as HiChIP [11] and SPRITE [12], and GAM can generate wholegenome contact maps, which can be used to reconstruct chromatin 3D architectures [13,14,15,16].
In the GAM experiment, a slice or an extra thin layer is cut at a random direction out of a single nucleus. By performing the same action on hundreds of nuclei at random orientations, GAM can gather hundreds of thin DNA layers. Each of these layers is referred as a nuclear profile (NP) that contains chromatin fragments that may be from different regions of a chromosome or even from different chromosomes [3]. After thin cryosections, five steps are performed in order to get the GAM DNA reads [3]: (1) fixing chromatins; (2) protein digestion and crosslink removal; (3) fragmentation (longer fragments are split into subfragments); (4) universal adaptor addition; and (5) amplification of the DNA reads. The DNA reads from the same NP have the same type of adaptor. Therefore, after being sequenced, all reads are mapped back to the reference genome to obtain their genomic locations.
After these steps, a cosegregation matrix is generated. Figure 1a, up shows three example NPs, each containing one or more bins (the orange, green, and blue colored beads in Fig. 1); and Fig. 1a, bottom shows the cosegregation matrix generated from these NPs. The number of columns in the cosegregation matrix is equal to the number of NPs, whereas the number of rows in the cosegregation matrix is equal to the number of bins of the whole genome. For example, the top three plus signs in the column of NP1 indicate that the bins A1, A2, and A3 are all detected in NP1. The top one plus sign in the column of NP2 indicates A1 is detected in NP2. However, A2 and A3 are not detected in NP2.
From the cosegregation matrix, the GAM contact matrix can be generated. The entries in the raw GAM contact matrices are obtained by using linkage disequilibrium D_{AB} = f_{AB} − f_{A}f_{B} [3], where f_{AB} is equal to the frequency of nuclear profiles in which two bins A and B are simultaneously detected; f_{A} or f_{B} is the frequency of nuclear profiles in which the locus A or B is detected. Therefore, it is possible that D_{AB} is less than zero, which makes the raw GAM contact matrix negative.
There are multiple types of systematic biases found in the raw population HiC data [17, 18] including the biases caused by different distances between successive restriction sites, GC content, and mappability. We recently confirmed that these biases also existed in raw singlecell HiC data [19]. GAM raw data were also found to have the GC content and mappability biases as mentioned in the paper that introduced the GAM technique [3]. Moreover, that paper [3] has mentioned that raw GAM data also have the window (or bins) detection frequency bias [3] that will be defined later in Methods section.
Windows and bins are defined interchangeably in this study as chromatin segments with the same length; and the window/bin size is also referred as the resolution, which is used to evenly split an entire chromatin. In contrast, we define fragments as the chromatin segments that are cut out based on the restriction enzymes in HiC, for example, the two fragments (blue and green) in Fig. 1b. We also define fragments as the chromatin segments existing in each of the NPs in GAM, for example, the blue and green fragments in Fig. 1c. The relationship between a fragment and a window/bin is that a fragment is usually longer and may contain multiple bins.
However, the GAM paper [3] misses an important bias that is caused by different lengths of fragments, named by us hereafter as the fragment length bias. This bias is similar to the bias of distances between restriction sites in raw HiC data. These biases (window detection frequency and fragment length) need to be removed to ensure that the significant interactions found in contact matrices are not resulted from the systematic biases.
In this paper, we demonstrate the existence of the fragment length bias in raw GAM data caused by different fragment lengths from random slicing, which was not discussed in [3]. A software package named normGAM has been developed that contains five methods for normalizing raw GAM data including normalized linkage disequilibrium (NLD) [3], vanilla coverage (VC) [1], sequential component normalization (SCN) [20], iterative correction and eigenvector decomposition (ICE) [21], and KnightRuiz 2norm (KR2) [2, 22].
Materials and methods
GAM and hiC data
We downloaded the raw cosegregation GAM data for 408 nuclear profiles (NPs) from GEO (GSE64881) at two resolutions (i.e., 1 Mb and 30 kb). The raw intrachromosomal (cis) GAM contact matrices were generated using the cosegregation data as input.
We downloaded the raw HiC reads for mouse embryonic stem (mES) cells from GEO (GSE35156) and combined two HindIII replicates to generate the raw HiC contact matrices at 1 Mb resolution.
Definition of the systematics biases
GC content bias and mappability bias
The GC content and mappability biases in GAM are the same as in HiC [17, 18]. Therefore, detailed definitions will not be shown here. The GC content bias of a certain bin is the GC content of its DNA sequence. The mappability bias of a certain bin is generated in the same way as in scHiCNorm [19].
Window detection frequency bias
The existence of window detection frequency (WDF) bias in raw GAM data has been mentioned in [3] and is intuitively apparent because GAM cannot ensure that all of the bins are detected with the same frequency. Our analysis has proved the existence of this bias (data shown later), which indicates that the random orientations of the GAM cryosections cannot make all the bins to have the same chance to be detected in the NPs. For example, bin A1 is detected three times in Fig. 1a, A2 is detected two times, and A3 two times. Accordingly, we define their WDF biases as 3, 2, and 2.
Fragment length bias
The fragment length bias is based on similar idea as the window detection frequency bias. The window detection frequency bias is caused by the inconsistent frequencies for bins to be detected in the NPs, whereas the fragment length bias is also a type of bias for each bin but is caused by the inconsistent frequencies for fragments to be detected in the NPs. Our evaluation results will later prove that both two biases exist in raw GAM contact matrices.
For example, in Fig. 1a the fragment length for the blue chromatin segment is three bins based on NP1. For NP2, it is one bin, and three bins in terms of NP3. Therefore, we define the average fragment length of bin A1 as (3 + 1 + 3)/3 based on all of the three NPs. Similarly, we define the average fragment length of bin A2 as (3 + 0 + 3)/3 according to the three NPs. We use the average fragment lengths from all NPs as the fragment length bias of the bin. In our software, we omit the division of the number of NPs because this value is the same for all the bins. In this way, the fragment length biases of A1, A2, and A3 are 7, 6, and 6 in the example in Fig. 1.
The WDF bias and the newly found fragment length bias in the raw GAM data can be quantified by the ld_nld_cis function in our normGAM package. However, when a user is using the package to normalize raw GAM data, he/she does not need to quantify these biases in order to run the normalization algorithms. Instead, the algorithm can directly remove all the biases. The quantification of these biases is to prove the existence of the biases and to evaluate the performance of our biasremoving algorithms. If GC content and mappability biases at different resolutions are of interest, users can obtain the corresponding bias data for different reference genomes from our previous normalization method scHiCNorm [19].
Normalization methods
The normalization methods developed for raw HiC contact matrices (e.g., KR [2, 22] and HiCNorm [18]) may not be directly used for raw GAM data because (1) the input of KR algorithm needs to be a nonnegative symmetric matrix, but the raw GAM contact matrices may contain negative entries; (2) HiCNorm [18] and scHiCNorm [19] were designed for count data, but the raw GAM contacts were composed of real numbers. To address the first problem, we developed a new method KR2, the 2norm of KR, which was able to handle negative contact matrices.
The KnightRuiz (KR) [22] algorithm was designed to balance a matrix. Given a nonnegative symmetric matrix D, the algorithm tries to find a vector x such that
where diag(x) is a diagonal matrix converted from x, and e represents a vector of all ones.
Eq. (1) can be turned into:
We can obtain the iterative from eq. (2):
We can also use Newton’s method to get an alternative to eq. (3):
which can be rearranged as:
Let D_{k} = D + diag (x_{k})^{−1} diag (Dx_{k}), which is symmetric as D. The next step is to use innerouter iteration schemes with conjugate gradient method to solve eq. (5). When both sides of eq. (5) are premultiplied by diag(x_{k}), we can get:
where B_{k} = diag (x_{k})Ddiag(x_{k}) and y_{k + 1} = diag (x_{k})^{−1}x_{k + 1}. Eq. (6) can be solved using conjugate gradient iteration. More details about the initial values and stopping criteria can be found in [22]. The final balanced matrix can be obtained by diag(x)Ddiag(x). In summary, the original KR was designed for balancing a matrix in the 1norm. Here, we present KR2 for balancing a GAM contact matrix in the 2norm by taking the following three steps. We first conduct elementwise product by D_{2} = D · D and then run the original KR algorithm on D_{2} to get balanced matrix D_{3} = KR(D_{2}). Finally, we obtain the normalized matrix D_{norm} = D_{3} · sign (D), where sign(D) denotes signed allones matrix; and the entries’ signs are consistent with the corresponding entries in D.
The vanilla coverage (VC) [1], sequential component (SCN) [20], and iterative correction and eigenvector decomposition (ICE) [21] normalization methods share similarities in their methodologies. The details of the three methods are given below. Each normalized entry is calculated by D_{ij}/(e_{i}e_{j}), where D_{ij} is the entry in the raw contact matrix D.
As for VC, e_{i} is 1norm of the ith row; and e_{j} is 1norm of the jth column. As for SCN, e_{i} is 2norm of the ith row; and e_{j} is 2norm of the jth column. The original SCN also uses maximum iterations to reduce errors, but we have found that SCN performs better when conducting only one iteration on GAM data. Therefore, the normalization results from SCN in this work were all using one iteration.
As for ICE, e_{i} is 1norm of the ith row divided by its mean value over nonzero bins; and e_{j} is 1norm of the jth column divided by its mean value over nonzero bins. There are two stopping criteria in ICE including the maximum iteration and error tolerance. Based on our experience, it is difficult to achieve a predefined error tolerance (e.g., 1e6) for all chromosomes; and when we set a large value (e.g., 6000) to the maximum iteration the normalized values for several chromosomes are extremely irregular. Therefore, we set it to 100 in this work. If the error is less than 1e3, the algorithm will be terminated.
Normalized Linkage Disequilibrium (NLD) was implemented in the same way as described in [3]. The normalized D_{AB} is calculated by D_{AB}/D_{max}, and D_{max} is the theoretical maximum and defined as:
Evaluation of the normalization methods
For each of the four biases (WDF, fragment length, mappability, and GC content), we sorted the chromatin bins in the whole genome (chr1 to chr19) based on their biases, stratified the whole bins of a genome into 20 sets, and then calculated the average number of GAM contacts between each pair of the 20 sets, resulting in a 20by20 heat map. We did the same operations on both the raw and normalized GAM contact matrices. A good normalization method can make the 20by20 heat map as smooth as possible, that is, the normalized 20by20 heat map is mostly occupied by one color.
Another evaluation measure is the Pearson’s correlation between the GAM data (both before and after normalization) and the four known biases. A good normalization method can achieve a very low correlation (i.e., close to zero) in terms of each of the four biases.
Results and discussion
We found that the new fragment length bias did exist in the raw GAM data at 1 Mb and 30 kb resolutions based on 408 NPs. Figure 2 is for 1 Mb resolution and Fig. 3 for 30 kb resolution. Compared with the blue regions with lower values, the red regions along the diagonal indicate the existence of biases. We did both Students’ ttests and Wilcoxon singedrank tests on the two data sets of WDF and fragment length biases at 1 Mb resolution for each of 19 chromosomes. The pvalues from the two tests for all chromosomes are consistently lower than 1e10, indicating that the two biases (WDF and fragment length) are significantly different from each other. Together with WDF, mappability, and GC content, we have detected four biases in the raw GAM data.
We evaluated the performance of the five normalization methods at 1 Mb and 30 kb resolutions. The normalization results are shown in Figs. 2, 3, and 4. Compared with the method we newly designed (KR2), the normalization method NLD that was previously used in the GAM paper [3] was not able to effectively eliminate fragment length, WDF, mappability, or GC content biases at 1 Mb resolution although its performance was better at 30 kb resolution for the biases of WDF and fragment length. All the five methods eliminated the four known biases in varying degrees, with two methods (i.e., VC and KR2) performing much better than the others at both resolutions.
We calculated the Pearson’s correlation between GAM data (both raw and normalized) and each of the four biases (Fig. 4), which showed that VC, ICE, and KR2 performed better than the other two methods. Notice that a zero or closetozero correlation indicates complete or closetocomplete removal of the bias.
We found KR2normalized GAM data were more significantly correlated with the KRnormalized HiC data compared with the ICE or VC normalized GAM and HiC data (Fig. 5), which indicated that the KRbased normalization methods might be a better choice if keeping consistency between GAM and HiC data is of interest.
We tested whether the normalization procedures reduced or damaged the genomic interacting patterns by plotting the raw and normalized GAM contact matrices, see Fig. 6. From the six heat maps, we can observe that the five normalization methods do not compromise the interacting patterns found in the raw GAM data.
Moreover, we tested the normalization efficiency by crosschecking the normalized GAM contacts with the fluorescence in situ hybridization (FISH) data [23], that is, by comparing with the normalized FISHdetected distances between six pairs of loci: three from chromosome 2 and the other three from chromosome 11. We calculated the Spearman’s rank correlation coefficient between the FISHdetected distances and each of the average GAM contacts from raw, NLPnormalized VCnormalized, SCNnormalized, ICEnormalized, and KR2normalized matrices; and the results are − 0.2, − 0.26, − 0.54, − 0.31, − 0.54, and − 0.49, respectively. Therefore, we can conclude that the normalization procedures not only remove systematic biases, but also make the raw GAM data more consistent with the FISHdetected distances.
Conclusion
We found the fragment length bias, a new type of systematic bias in raw GAM data that has not been noticed before in literature. We proved that this bias existed in raw GAM data at both 1 Mb and 30 kb resolutions. We designed a new normalization method (i.e., KR2) to remove the four known biases in raw GAM data and implemented other four widelyused normalization methods in R, including NLD, VC, SCN, and ICE. We evaluated five normalization methods (i.e., NLD, VC, SCN, ICE, and KR2) at 1 Mb and 30 kb resolutions. Our evaluation results show that the five normalization methods can remove, to different extents, the four known biases; and two of them (i.e., VC and KR2) perform better than the rest. Compared with the other three methods (i.e., NLD, VC, and ICE), the KR2normalized GAM data are more consistent with the KRnormalized HiC data. We also showed that the normalized GAM data (biases removed) had a higher correlation with FISH data compared with raw GAM data.
Abbreviations
 FISH:

fluorescence in situ hybridization
 GAM:

genome architecture mapping
 ICE:

iterative correction and eigenvector decomposition
 KR2:

KnightRuiz 2norm
 NLD:

normalized linkage disequilibrium
VC
vanilla coverage
 NP:

nuclear profile
 SCN:

sequential component normalization
References
 1.
LiebermanAiden E, van Berkum NL, Williams L, Imakaev M, Ragoczy T, Telling A, Amit I, Lajoie BR, Sabo PJ, Dorschner MO, et al. Comprehensive mapping of longrange interactions reveals folding principles of the human genome. Science. 2009;326(5950):289–93.
 2.
Rao SS, Huntley MH, Durand NC, Stamenova EK, Bochkov ID, Robinson JT, Sanborn AL, Machol I, Omer AD, Lander ES. A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping. Cell. 2014;159(7):1665–80.
 3.
Beagrie RA, Scialdone A, Schueler M, Kraemer DC, Chotalia M, Xie SQ, Barbieri M, de Santiago I, Lavitas LM, Branco MR. Complex multienhancer contacts captured by genome architecture mapping. Nature. 2017;543(7646):519.
 4.
Wang Z, Cao R, Taylor K, Briley A, Caldwell C, Cheng J. The properties of genome conformation and spatial gene interaction and regulation networks of normal and malignant human cell types. PLoS One. 2013;8(3):e58793.
 5.
Nagano T, Lubling Y, Stevens TJ, Schoenfelder S, Yaffe E, Dean W, Laue ED, Tanay A, Fraser P. Singlecell hiC reveals celltocell variability in chromosome structure. Nature. 2013;502(7469):59–64.
 6.
Ramani V, Deng X, Qiu R, Gunderson KL, Steemers FJ, Disteche CM, Noble WS, Duan Z, Shendure J. Massively multiplex singlecell hiC. Nat Methods. 2017;14(3):263–6.
 7.
Zhu H, Wang Z. SCL: a latticebased approach to infer 3D chromosome structures from singlecell HiC data. Bioinformatics. 2019;35(20):3981–88.
 8.
Dekker J, Belmont AS, Guttman M, Leshyk VO, Lis JT, Lomvardas S, Mirny LA, O’shea CC, Park PJ, Ren B. The 4D nucleome project. Nature. 2017;549(7671):219.
 9.
Wang Y, Song F, Zhang B, Zhang L, Xu J, Kuang D, Li D, Choudhary MN, Li Y, Hu M. The 3D genome browser: a webbased browser for visualizing 3D genome organization and longrange chromatin interactions. Genome Biol. 2018;19(1):151.
 10.
Segal MR, Bengtsson HL. Improved accuracy assessment for 3D genome reconstructions. BMC Bioinformatics. 2018;19(1):196.
 11.
Mumbach MR, Rubin AJ, Flynn RA, Dai C, Khavari PA, Greenleaf WJ, Chang HY. HiChIP: efficient and sensitive analysis of proteindirected genome architecture. Nat Methods. 2016;13(11):919.
 12.
Quinodoz SA, Ollikainen N, Tabak B, Palla A, Schmidt JM, Detmar E, Lai MM, Shishkin AA, Bhat P, Takei Y, et al. Higherorder interchromosomal hubs shape 3D genome Organization in the Nucleus. Cell. 2018;174(3):744–57 e724.
 13.
Liu T, Wang Z. Reconstructing highresolution chromosome threedimensional structures by hiC complex networks. BMC Bioinformatics. 2018;19(Suppl 17):496.
 14.
Zou C, Zhang Y, Ouyang Z. HSA: integrating multitrack hiC data for genomescale reconstruction of 3D chromatin structure. Genome Biol. 2016;17(1):1.
 15.
Varoquaux N, Ay F, Noble WS, Vert JP. A statistical approach for inferring the 3D structure of the genome. Bioinformatics. 2014;30(12):i26–33.
 16.
Hu M, Deng K, Qin Z, Dixon J, Selvaraj S, Fang J, Ren B, Liu JS. Bayesian inference of spatial organizations of chromosomes. PLoS Comput Biol. 2013;9(1):e1002893.
 17.
Yaffe E, Tanay A. Probabilistic modeling of hiC contact maps eliminates systematic biases to characterize global chromosomal architecture. Nat Genet. 2011;43(11):1059–65.
 18.
Hu M, Deng K, Selvaraj S, Qin Z, Ren B, Liu JS. HiCNorm: removing biases in hiC data via Poisson regression. Bioinformatics. 2012;28(23):3131–3.
 19.
Liu T, Wang Z. scHiCNorm: a software package to eliminate systematic biases in singlecell hiC data. Bioinformatics. 2018;34(6):1046–7.
 20.
Cournac A, MarieNelly H, Marbouty M, Koszul R, Mozziconacci J. Normalization of a chromosomal contact map. BMC Genomics. 2012;13(1):436.
 21.
Imakaev M, Fudenberg G, McCord RP, Naumova N, Goloborodko A, Lajoie BR, Dekker J, Mirny LA. Iterative correction of hiC data reveals hallmarks of chromosome organization. Nat Methods. 2012;9(10):999–1003.
 22.
Knight PA, Ruiz D. A fast algorithm for matrix balancing. IMA J Numer Anal. 2013;33(3):1029–47.
 23.
Eskeland R, Leeb M, Grimes GR, Kress C, Boyle S, Sproul D, Gilbert N, Fan Y, Skoultchi AI, Wutz A. Ring1B compacts chromatin structure and represses gene expression independent of histone ubiquitination. Mol Cell. 2010;38(3):452–64.
Acknowledgements
Publication of this article was sponsored by the National Institutes of Health R15GM120650 grant to ZW and startup funding from the University of Miami to ZW.
About this supplement
This article has been published as part of BMC Genomics Volume 20 Supplement 12, 2019: The International Conference on Intelligent Biology and Medicine (ICIBM) 2019: Bioinformatics methods and applications for human diseases: genomics. The full contents of the supplement are available online at https://bmcgenomics.biomedcentral.com/articles/supplements/volume20supplement12 .
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Both authors have read and approved the final manuscript. Methodology, TL; Software, TL; Evaluation, TL; Visualization, TL; Investigation, TL and ZW; Writing – Original Draft Preparation, TL; Writing – Review & Editing, ZW; Supervision, ZW; Project Administration, ZW; Funding Acquisition, ZW.
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Correspondence to Zheng Wang.
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Liu, T., Wang, Z. normGAM: an R package to remove systematic biases in genome architecture mapping data. BMC Genomics 20, 1006 (2019) doi:10.1186/s1286401963318
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Keywords
 Genome architecture mapping
 GAM
 HiC
 3D genome structure
 Normalization
 Systematic biases