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FRONTIERS IN ONCOLOGY


CANCER GENETICS

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THIS ARTICLE IS PART OF THE RESEARCH TOPIC

Advances in Mathematical and Computational Oncology, Volume II View all 11
Articles

Articles

EDITED BY

MARIA RODRIGUEZ MARTINEZ



IBM Research - Zurich, Switzerland

REVIEWED BY

KIMBERLY GLASS



Brigham and Women's Hospital, Harvard Medical School, United States

SHENG LIU



Indiana University school of medicine, United States

NOEMI EIRO



Jove Hospital Foundation, Spain

The editor and reviewers' affiliations are the latest provided on their Loop
research profiles and may not reflect their situation at the time of review.

TABLE OF CONTENTS

 * * Abstract
   * 1 Introduction
   * 2 Materials and Methods
   * 3 Results
   * 4 Discussion
   * Data Availability Statement
   * Author Contributions
   * Funding
   * Conflict of Interest
   * Publisher’s Note
   * Acknowledgments
   * Supplementary Material
   * References




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Open Supplemental Data


ORIGINAL RESEARCH ARTICLE

Front. Oncol., 17 November 2021 | https://doi.org/10.3389/fonc.2021.726493


GENE CO-EXPRESSION IN BREAST CANCER: A MATTER OF DISTANCE

Alfredo González-Espinoza1,2, Jose Zamora-Fuentes2, Enrique Hernández-Lemus2,3
and Jesús Espinal-Enríquez2,3*
 * 1Department of Biology, University of Pennsylvania, Philadelphia, PA, United
   States
 * 2Computational Genomics Division, National Institute of Genomic Medicine,
   Mexico City, Mexico
 * 3Centro de Ciencias de la Complejidad, Universidad Nacional Autόnoma de
   México, Mexico City, Mexico

Gene regulatory and signaling phenomena are known to be relevant players
underlying the establishment of cellular phenotypes. It is also known that such
regulatory programs are disrupted in cancer, leading to the onset and
development of malignant phenotypes. Gene co-expression matrices have allowed us
to compare and analyze complex phenotypes such as breast cancer (BrCa) and their
control counterparts. Global co-expression patterns have revealed, for instance,
that the highest gene-gene co-expression interactions often occur between genes
from the same chromosome (cis-), meanwhile inter-chromosome (trans-)
interactions are scarce and have lower correlation values. Furthermore, strength
of cis- correlations have been shown to decay with the chromosome distance of
gene couples. Despite this loss of long-distance co-expression has been clearly
identified, it has been observed only in a small fraction of the whole
co-expression landscape, namely the most significant interactions. For that
reason, an approach that takes into account the whole interaction set results
appealing. In this work, we developed a hybrid method to analyze
whole-chromosome Pearson correlation matrices for the four BrCa subtypes
(Luminal A, Luminal B, HER2+ and Basal), as well as adjacent normal breast
tissue derived matrices. We implemented a systematic method for clustering gene
couples, by using eigenvalue spectral decomposition and the k–medoids algorithm,
allowing us to determine a number of clusters without removing any interaction.
With this method we compared, for each chromosome in the five phenotypes: a)
Whether or not the gene-gene co-expression decays with the distance in the
breast cancer subtypes b) the chromosome location of cis- clusters of gene
couples, and c) whether or not the loss of long-distance co-expression is
observed in the whole range of interactions. We found that in the correlation
matrix for the control phenotype, positive and negative Pearson correlations
deviate from a random null model independently of the distance between couples.
Conversely, for all BrCa subtypes, in all chromosomes, positive correlations
decay with distance, and negative correlations do not differ from the null
model. We also found that BrCa clusters are distance-dependent, meanwhile for
the control phenotype, chromosome location does not determine the clustering. To
our knowledge, this is the first time that a dependence on distance is reported
for gene clusters in breast cancer. Since this method uses the whole cis-
interaction geneset, combination with other -omics approaches may provide
further evidence to understand in a more integrative fashion, the mechanisms
that disrupt gene regulation in cancer.




1 INTRODUCTION


1.1 BREAST CANCER: A COMPLEX DISEASE

Breast cancer is the first cancer-related cause of death in women worldwide. It
is also, according to the most recent data (1), the most diagnosed neoplasm in
the world. Breast cancer is also the malignant neoplasm with the highest
incidence (1). Its diagnosis, response to treatment, relapse, and outcome are
strongly determined by the molecular profile underlying the disease (2–4). The
PAM50 classifier is among the most relevant methods of classification for breast
cancer molecular subtypes (5). This molecular classification is based on the
expression signature of 50 genes relevant to the oncogenic phenotype (5–7).

Publicly available massive cohorts of genomic and clinical data in the study of
cancer, have allowed the analysis of an immeasurable amount of information. The
latter has contributed to a better understanding of the oncogenic process (8).
Based on gene expression of hundreds-to-thousands of samples, now it is possible
to study such vast experimental information to infer and analyze the
whole-genome co-expression landscape, aiming to highlight similarities and
differences between cancer and non-cancer samples. Among these efforts, The
Cancer Genome Atlas (TCGA) has contributed in an outstanding way (9).


1.2 GENE CO-EXPRESSION NETWORKS

The study of Cancer within the framework of complex networks has become
increasingly relevant in the last years (10–20). Given its size and complexity,
genome-wide regulation may include a large number of features (all the genes),
potentially inducing a fully connected network, with contributions of very
different relevance and certainty. For this reason, several approaches to reduce
its dimensionality have been implemented, including the use of threshold
methods, to look for the most significant co-expression relationships (18, 21).
In particular, in the case of breast cancer molecular subtype networks, the most
significant co-expressed pairs have been used as connected nodes in biologically
relevant modules (22–25).

Further approaches to determine the optimal network size may analyze a wide
range of network scales (13, 26, 27) or backbone-related threshold networks
(28), and even use gene co-expression subsets of clinical/biological relevance
(29).

In the attempt of reducing the dimensionality of a fully-connected network,
identification of groups of genes that behave in a similar way –indicating that
their expression profiles are correlated– is a relevant problem and is still an
open challenge in network biology (29, 30). The latter point is closely related
to the so-called graph sparsification problem in graph theory. The choose of a
significance threshold then becomes relevant.

For instance, in a recent study by Kimura et al. (31), an approach was developed
to select parameters in genetic networks by computational methods (mainly
Machine Learning and Artificial Intelligence). Other approaches have used the
complete set of interactions in order to construct a network backbone (28).
There, the authors used the complete matrix of interactions to obtain the most
important relationships, preserving those edges with statistically significant
deviations with respect to a null model for the local edge’s weight assignment.


1.3 GENE CO-EXPRESSION IS DISTANCE DEPENDENT

In cancer, gene co-expression networks have been used to uncover genes and
relationships that may represent crucial elements to determine differences
between phenotypes (32). In particular, in breast cancer and breast cancer
molecular subtypes (4), gene co-expression networks have been useful to identify
the phenomenon of loss of long-range co-expression (10, 12, 14, 33): this is, a
property observed in cancer networks in which the most significant gene
co-expression relationships occur between genes that belong to the same
chromosome, i.e., cis- interactions. Conversely, inter-chromosome (trans-)
interactions are often weak in cancer.

Furthermore, the loss of long-range co-expression is not only observed at the
level of genes located on different chromosomes. Regarding cis-
(intra-chromosome) gene interactions, there is an exponential decay of strength
of correlations (14) as genes become more distant. This situation could be
related to a diminishing of the accessibility that a certain region of the
genome may have of its environment during the carcinogenic process. Importantly,
this lack of accessibility can be attributed to several factors, among which we
can mention aberrant expression of transcription factors, copy number
alterations, incorrect binding to CTCF, or changes in Topologically Associated
Domains (TADs). All of these factors have the potential to alter, both, the
structure of DNA and gene expression.

Despite this phenomenon has been discovered not only in breast cancer, but also
in clear cell renal carcinoma (13), lung adenocarcinoma and squamous cell lung
carcinoma (12), loss of long-range co-expression has been determined for the top
highest interactions: a small subset of the most co-expressed gene-gene
interactions (tens-to-hundreds of thousands) of the whole co-expression
landscape is observed to be biased to cis- interactions.

Since the strength of intra-chromosome interactions have been observed to be the
highest ones, it becomes important to evaluate the behavior of the whole
intra-chromosome landscape of cancer networks. In these terms, network
clustering may provide us with information related to, for example, sets of
genes constrained by physical restrictions in certain regions of the genome,
genes that act in tandem, events related with the transcriptional process, etc.

To address the questions above, we performed a data-driven clustering analysis
using a hybrid algorithm that involves eigenvalue decomposition and k–medoids
from correlation matrices of each chromosome. These matrices were inferred from
RNA-Seq-based gene expression. We evaluated whether or not the loss of
long-range co-expression is preserved, by studying all chromosomes for the four
breast cancer subtypes as compared with normal tumor-adjacent tissue as control.

With this approach, we constructed co-expression matrices for all chromosomes in
adjacent normal breast tissue network, as well as in all four breast cancer
subtypes. We analyzed the statistics for their clustering nearest neighbor
distributions within each chromosome, comparing each breast cancer molecular
subtype as well as the adjacent normal tissue. Additionally, for all phenotypes,
we constructed a null model to provide statistical robustness to our analyses.
With this, we present a systematic method for intra-chromosome gene clustering,
which allows to compare the whole co-expression landscape between a cancerous
phenotype with its control counterpart.


2 MATERIALS AND METHODS


2.1 DATA ACQUISITION

Gene expression data of breast invasive carcinoma was collected from The Cancer
Genome Atlas (TCGA) (34). 735 tumor and 113 non-cancerous (adjacent normal),
samples were considered, see Table 1. Illumina HiSeq RNASeq samples were
filtered (biotype, expression mean >10), pre-processed, and log2 normalized gene
expression values as described in (10). We performed data corrections for
transcript length, GC content and RNA composition. Tumor expression values were
classified using PAM50 algorithm into the respective intrinsic breast cancer
sub-types (Luminal A, Luminal B, Basal, and HER2-Enriched) using the
Permutation-Based Confidence for Molecular Classification (35) as implemented in
the pbcmc R package (36).


TABLE 1

Table 1 Samples for each subtype.



Tumor samples with a non-reliable breast cancer sub-type call were removed from
the analysis. To avoid overlapping patterns among subtype expression values,
multidimensional noise reduction was performed using ARSyN R implementation
(37), and a multidimensional Principal Component Analysis (PCA) was implemented
to confirm noise reduction (14).

Since a crucial part of this work lies in having a highly-confident set of
matrices, it is necessary to obtain as many well-characterized samples as
possible, for each molecular subtype. Due to this fact, we decided to include
all the available samples with a molecular subtype classification i.e., those
samples with a molecular subtype label from the original source. Further
investigations must be conducted with even more stringent inclusion and
exclusion criteria, such as histologically confirmed diagnosis,
histopathologically-assessed axillary lymph nodes, metastatic disease at
presentation, adjuvant treatment, etc.

In order to provide all the information to reproduce our results, the clinical
information about histological data by subtype-samples is now included in the
Supplementary Material S1. There, for each breast cancer subtype sample we
describe: 1) availability of historical adjuvant treatment, 2) lymph node
assessment existence, 3) histological type of tumor and 4) axillary
lymph-node-stage method type.

To show that those samples with the same molecular subtype are indeed properly
classified in their molecular profiles to be included in our correlation
matrices, we performed a Principal Component Analysis (PCA) for each subtype
(Supplementary Figure S1). The PCA groups samples based on the main eigenvalues
of the expression profiles. In this case, we present the two main principal
components (X and Y axes of the Supplementary Figure S1) -though the
calculations were made with the full eigenvalue spectra of the matrices. Hence,
the PCA could indicate those samples that are not similar to the rest of their
class (if any) or if there is any “confounded” or misclassified sample.

As it can be noticed in the Supplementary Figure S1, all subtype samples are
clearly separated based on the molecular classification. All samples are grouped
by its subtype (color). Hence, constructing correlation matrices by using these
subtype-separated samples, certainly improves the statistical significance
without adding a clear source of noise.


2.2 CORRELATION MATRICES

We built intra-chromosomal cross-correlation matrices by estimating the Pearson
correlation coefficient between the expression of two genes i and j, defined as
follows:

Cij=Cov(gi,gj)σgiσgj=1Ns∑s=1Ns(gis−μgi)(gjs−μgj)σgiσgj,(1)Cij=Cov(gi,gj)σgiσgj=1Ns∑s=1Ns(gis−μgi)(gjs−μgj)σgiσgj,(1)


where gi is the set of Ns expression samples for gene i. By definition, a
correlation matrix is symmetric (Cij = Cji), the elements in the diagonal are 1
(Cii = 1, ∀i), and its values are bounded to –1 ≤ Cij ≤ 1, where Cij = 1
corresponds to perfect correlations, Cij = –1 corresponds to perfect
anticorrelations, and Cij = 0 corresponds to uncorrelated gene pairs.

We calculated Pearson correlation between all genes for each chromosome for the
five phenotypes. The code for calculation of Pearson correlations can be found
in (38).


2.3 SPECTRAL DECOMPOSITION

Pearson correlation matrices for each chromosome were calculated in order to
analyze their spectral properties. Previous works on correlation matrices have
shown that their spectral properties carry information about the structure and
dynamics of the system (39–48).

For example, in stock market data, the first eigenvectors correspond to clusters
of related industries (49, 50). In Electroencephalography measurements, these
eigenvectors correspond to different functional regions in the brain (51).
However, not all of the eigenvalues carry relevant information about the system.
It has been shown that the smallest ones are the most sensitive to noise and
some of them correspond to weak interrelations between small components from
different clusters (47, 48). To distinguish how many eigenvalues contain useful
information to identify clusters, we compared the spectral properties of the
empirical correlation matrix to a null model represented by an ensemble of
random matrices.

This ensemble of random matrices, is obtained by doing non-biased shuffling over
the gene expression values for each sample (in this way, the original
distribution of the data is preserved while its correlations will be destroyed)
and computing the correlation matrix of each randomized data as in equation 1,
we generated an ensemble of nm = 100 random matrices for each chromosome and
phenotype.

The k deviating eigenvalues of the empirical matrix from the randomized data
max(λR) < {λ1, … ,λk} are the ones containing correlations that cannot be
attributed to either the noise in the system or data randomization. It is worth
noticing that instead of using the eigenvectors from the spectral decomposition,
which can be difficult to separate into independent clusters (52) (see
Supplementary Figure S2), we used the number of k deviating eigenvalues as the
number of independent clusters for a different clustering method.


2.4 CLUSTERING ANALYSIS

We implemented a clustering analysis based on the k-medoids algorithm. In a
similar fashion to k-means, k-medoids clustering attempts to minimize the
distance between the elements inside a cluster but one element is designated as
the center of the cluster. The k-medoids algorithm works not exclusively with
Euclidean distances, but with general pairwise interactions, this means we can
use the correlation values we have estimated for each intra-chromosome matrix.
Since correlation values are signed and their magnitude goes from –1 to 1, we
define the pairwise interactions between genes i and j as:

Di,j≡1−|Ci,j|,(2)Di,j≡1−|Ci,j|,(2)


with 0 ≤ D ≤ 1, high correlation or anti-correlation values mean close distance
between points, while small correlation values will give higher distances.
Finally, for the parameter k in the clustering algorithm, we considered the
number of deviating eigenvalues as obtained from the spectral decomposition.

Given the stochastic nature of the k-medoids algorithm, we did nr = 100
realizations for each clustering computation to ensure statistical significance
(p < 0.01), choosing the output configuration as the one with the minimum mean
distance between the centroids and the elements in each cluster.

In order to compare the clustering results between the control phenotype and any
other cancer subtype in a given chromosome, we constructed the intra-cluster
Nearest Neighbor Distance (NND) distribution for each subtype. The NND of a
given gene i in a cluster k is defined as:

Dinn≡min(∣∣j−i∣∣)∀j∈Ck,(3)Dnni≡min(|j−i|)∀j∈Ck,(3)


where Ck refers to the cluster k. To quantify the difference between the
clustering in adjacent normal and cancer subtypes we compute Shannon’s entropy
H(x)=−Σx∈χp(x)log(p(x))H(x)=−Σx∈χp(x)log(p(x)) for the NND distributions, which
in this case can be interpreted as how localized or how spread are the genes
within each cluster in the chromosome. We also computed the Kolmogorov-Smirnov
distance between the adjacent normal case and each of the Cancer subtypes. Given
two cumulative distribution functions (CDF) the Kolmogorov-Smirnov distance is
defined as:

DKS(Fn,Fm)=supx∣∣∣Fn(x)−Fm(x)∣∣∣,(4)DKS(Fn,Fm)=supx|Fn(x)−Fm(x)|,(4)


where the functions Fn and Fm are the CDFs for two samples n and m.


3 RESULTS

A correlation matrix of the sort just described, can be visualized as a heatmap
as shown in Figure 1 where correlation matrices for adjacent normal and basal
subtype samples in the chromosome 1 are displayed. The axis represent the genes
ordered by their physical location in the chromosome. The clearest difference
between both matrices seems to be the lowest value of absolute correlation for
genes that are physically distant in the basal subtype case. The heatmaps for
each chromosome in the five phenotypes can be observed in Supplementary
Materials S2–S6.


FIGURE 1

Figure 1 Pearson correlation square matrices for chromosome 1 in control samples
(up) and basal breast cancer subtype (down). Genes are placed according to their
physical location on the chromosome. Colors represent the correlation value: red
corresponds to positive values, meanwhile negative correlations are depicted in
blue. The inserts in the right part of both matrices correspond to the
scatterplot of Pearson correlations versus distance. The horizontal red line
corresponds to Pearson correlation = 0.



The effect of loss in long range co-expression is consistent with previous works
of regulatory networks in breast cancer (10, 12–14, 33, 53). The block-type
structure of the basal subtype matrix suggests the utility of clustering
analysis to compare the structural properties of the correlation matrices. In
what follows, we will present results for these clustering analyses. Through the
manuscript, the presented figures will show different chromosomes for the five
phenotypes. This has been done, in order to illustrate the universal nature of
the gene clustering in breast cancer molecular subtypes, compared with the
adjacent normal tissue.


3.1 CO-EXPRESSION DECAYS IN ALL CHROMOSOMES IN ALL SUBTYPES

We observed a common pattern of distance dependency in all chromosomes in all
breast cancer phenotypes. The decay in gene co-expression corresponds
exclusively to positive correlations. In the case of negative correlations, such
effect is not observed. Conversely, in adjacent normal chromosomes, there is no
dependency of distance neither in negative nor positive interactions.
Interestingly, this effect is observed in all chromosomes in the four breast
cancer molecular subtypes and not observed in adjacent normal breast
tissue-derived correlations (Figure 2 and Supplementary Materials S7–S11).


FIGURE 2

Figure 2 Pearson correlation of gene-gene expression versus distance. Plots for
adjacent normal and cancer subtypes of chromosome 8 (green) and their respective
null model (orange). The solid lines represent the median of a moving average in
the distribution of correlation values over each window and the shaded area is
the range from its first and third quartiles.



In order to evaluate the differences between the empirical data and the null
model, we performed a non-parametric hypothesis test (Kolmogorov-Smirnov) for
the correlation values distributions (in all tumor subtypes and adjacent normal
tissue) versus phenotype-specific null models. Additionally we implemented their
corresponding significance tests (obtained via bootstrap/permutation analysis).
The results of the KS test can be observed in Figure 3. The results for the rest
of chromosomes, as well as their significance p-values, are presented in
Supplementary Materials S12, S13).


FIGURE 3

Figure 3 KS hypothesis test between empirical data from chromosome 8 and null
model for the five phenotypes. This plot represents the KS statistic versus
distance for all phenotypes in chromosome 8. The p– value for the control
phenotype is smaller than 10–5.



Notice that at short distances, the cancer phenotypes have larger values than
the adjacent normal correlations. However, at larger distances, KS for adjacent
normal network are larger than those for cancerous phenotypes. The p-values
shown in the upper right part of the figure, represent the average of all set of
distances.

Based on a null model that lacks the linear correlations from the original data
(see Methods), we observed that in adjacent normal chromosomes, positive and
negative correlation values seem to be independent of the distance between
genes, having significantly higher absolute values when compared with the null
model at any distance. In the case of cancer subtypes, negative correlations are
non-significant, but a few small regions in specific chromosomes (See
Supplementary Figure S3).


3.2 EIGENVALUE DECOMPOSITION DEFINES THE NUMBER OF CLUSTERS IN ALL PHENOTYPES

We generated an ensemble of (N = 100) random matrices and compare the eigenvalue
distributions from both, original and random matrices (see Materials and
Methods). The left panel of Figure 4 shows the eigenvalue distribution for the
ensemble of random matrices, where its shape is the well-known Marchenko-Pastur
distribution from random matrix theory (54). Overlapped eigenvalue distributions
for the original matrix of chromosome 17 and the ensemble of surrogates are
shown in the right panel of Figure 4. A set of significant eigenvalues was
determined by random matrix permutations (p < 0.01) (see box in Figure 4).


FIGURE 4

Figure 4 Probability distributions of eigenvalues for a) the ensemble of random
matrices, b) random and empirical data for the chromosome 17 in the Basal
sub-type.




3.3 GENE CLUSTERING IS DISTANCE DEPENDENT IN BREAST CANCER

With the method referred in Section 3.2 we obtained the full set of clusters for
each chromosome in all phenotypes. Figure 5 shows Chromosome 19 clusters with
genes sorted by gene start base pair position. In the adjacent normal chr19
figure (upper part) we cannot discern a pattern in cluster colors. The
distribution of clusters does not seem to depend on the distance between genes.
Meanwhile, in basal breast cancer, we can observe cluster panels of colors,
clearly detached. In the same figure, in the right panels, we plot the
cumulative distribution of genes for each cluster. The larger the slope, the
more often contiguous genes belong to the same cluster. All clusters for the
five phenotypes in all chromosomes can be found in Supplementary Material S14.
Cumulative distribution for all clusters can be found in Supplementary Material
S15.


FIGURE 5

Figure 5 Cluster assembly of Chromosome 19 for adjacent normal-tissue and Basal
breast cancer matrices. Upper part: Heatmap for all clusters in the adjacent
normal matrix. Lower part: analog heatmap for Basal breast cancer network. The
right panels correspond to the cumulative distribution for each cluster in
chromosome 19. Colors represent the top-10 largest clusters.



Cumulative distributions for the Nearest Neighbor Distance (NND) of two
different chromosomes are shown in Figure 6, which can be interpreted as the
probability distribution of the minimum distance between two genes in the same
cluster. The behavior seen in the previous Figure 5 holds: genes from the same
cluster are more likely to be close to each other.


FIGURE 6

Figure 6 Cumulative NND distributions for each subtype in Chromosomes 14 and 19.
Distributions of Cancer subtypes have different behavior in short and long
distances compared with the adjacent normal tissue.



Results for the entropies for the NND distributions are shown on the left panel
of Figure 7, where a clear trend with the value H(x) can be identified: Luminal
A, Luminal B, HER2+, Basal. It is worth noticing that the aforementioned order
coincides with survival rates and metastatic behavior (14, 55, 56). The subtypes
with the lowest survival rates and more metastatic behavior also present lower
entropy values.


FIGURE 7

Figure 7 (A) Entropy for the NND distribution in all chromosomes. (B)
Kolmogorov-Smirnov distance between the CDF of the NND in the adjacent normal
phenotype and each of the Cancer subtypes for each chromosome.



The latter is in agreement with a previously observed trend for the top 0.1%
gene co-expression interactions for the four phenotypes: The most aggressive
phenotype (basal) has the lowest number of inter-chromosome interactions,
meanwhile the Luminal A subtype, which is considered the one with the best
prognosis, contains a much larger fraction of interactions between genes from
different chromosomes (14).

The decay in the entropy for the NND distribution presents further evidence that
in the cancer subtypes, genes co-express in tighter patterns, in contrast with
genes in the control phenotype that co-express at broadly scattered distances
over the chromosome. A similar trend holds for the KS distance between the
control phenotype and each subtype in the right panel of Figure 7, where higher
values indicate a larger difference in the spatial organization of the clusters.
The difference in spatial organization within the clusters in the chromosome is
evident with both measures and it is correlated with the survival rate and
metastasis of the subtype (14).


4 DISCUSSION

Cancer research increasingly requires comprehensive computational analysis
tools. In the search for relevant biological information, it is essential to be
able to find selective patterns of individual or collective gene expression. In
this sense, clustering methods are becoming a pivotal computational tool.

In this work, we studied the co-expression of genes in breast cancer molecular
subtypes. We implemented a method to find the optimal clustering between genes
that are co-expressed. We observed a grouping pattern in the case of cancer
phenotypes with respect to the adjacent normal group. These patterns in the
genome indicate that in cancer, physically close genes are co-expressed (cis-
interactions), while for distant genes (trans- interactions) the clustered
co-expression is, to a large extent, lost.

The piece-wise Kolmogorov-Smirnov tests for all tumor subtypes and adjacent
normal tissue versus phenotype-specific null models and their corresponding
significance tests (obtained via bootstrap/Permutation analysis) (included in
the Supplementary Materials S12, S13), show that correlation values at short
distances are much more significant for all chromosomes in any cancer phenotype
than the adjacent normal network.

It is worth noticing that the significance of the KS tests also decays with the
distance for all chromosomes in any cancer phenotype. Conversely, for the
adjacent normal network, distance does not exert a considerable influence in the
significance of the KS test. Finally, the KS test also show that the
significance of differences between the correlation of our empirical data with
the null model is unique for each chromosome and for each phenotype.

The fact that genes are highly co-expressed in groups with close positions, may
be due to a favored number of nearby transcription sites or to the strong
presence of transcription factors. It has been observed for instance, that in
Luminal A breast cancer gene co-expression networks, co-factors, CTCF binding
sites (57, 58) or copy number alterations (59, 60), may remodel chromatin making
it more or less accessible, thus allowing gene transcription of local
neighborhoods, resulting in the concomitant high co-expression between those
neighboring genes (53). On the other hand, TFs influence more often
inter-chromosome edges, meanwhile intra-chromosome interactions are less
affected by them (53).

Physical interactions such as CTCF binding sites have captured attention in
recent years (61, 62), and more importantly, in breast cancer (63).

For instance, in (53), we constructed an intra-chromosome gene-co-expression
network for Luminal A breast cancer samples. There, a community detection method
was performed to determine whether CTCF binding sites appeared in the borders of
those communities. We observed that there is no link between CTCF binding sites
and the border of intra-chromosome communities. In that sense, we argued that,
at least for Luminal A breast cancer gene co-expression networks, CTCF binding
sites are not determinant for network structure.

Transcription factor (TF) regulation is, of course, one of the central
mechanisms for gene regulation. With respect to the role that TFs may exert on
gene clustering, we have previously shown that TFs influence genes in a trans-
fashion, i.e. TFs from a given chromosome regulate genes from different
chromosomes. We have shown that in terms of Master Regulators in breast cancer
(64, 65), but also in Luminal A breast cancer networks (53). Conversely, for
intra-chromosome genes, TFs influence is much less evident.

Finally, Copy Number Variations (CNVs) have been considered as a crucial factor
in the rise and development of breast cancer (59). In fact, a correlation
between CNVs, protein levels and mRNA gene expression has also been reported
previously (66). Hence, high correlations between clusters of physically closed
genes appear to be related to copy number alterations.

We have used TCGA-derived CNV data and compared the amplification/deletion peaks
with LumA network communities. Interestingly, the community with more
overexpressed genes, composed of genes such as FOXM1, HJURP, or CENPA, presented
large regions of deletions. The apparently contradictory result suggested that
the copy number alterations do not influence the structure of that community. On
the other hand, a gene community formed by HLA family genes, presented a common
pattern of amplification, but those genes were not differentially expressed
(53).

With the aforementioned in mind, we argue that CNVs are not as relevant as one
could expect in terms of the gene clustering shown here. Moreover, CNVs
influence may be at the expression level, but said effect is more limited
regarding co-expression. However, further investigation is necessary to clarify
these ideas.

The structure of clustered genes in physically close neighborhoods resembles the
images obtained by Hi-C methods (67–69).

In recent times, there has been an increased interest as to how chromosome
conformation capture experiments such as Hi-C may lead to relevant clues towards
our understanding of further effects in connection with transcriptional
regulation. Indeed, we are currently conducting research along these lines in
our group. Work is ongoing, however, we can advance that there seems to be
important correlations between loss/gain of statistically significant
chromosomal contacts and co-expression relationships between genes in the
associated genomic regions. It remains to be determined however whether said
correlations are significant via proper assessment of null models and, more
importantly, to determine what may be the biological consequences of these
associations.

Preliminary findings from our Hi-C analysis in breast cancer indicate that more
relevant contacts are mostly (but not exclusively) on close genomic regions.
This is not unlike what we have observed with MI-based gene co-expression
networks in which there is a preponderance of co-expression interactions in
shorter distances for tumors. Future work undoubtedly will focus on the
comparison between the network clusters constructed by this method and those
from Hi-C. In particular, the zones/genes between gene groups. The assessment
and comparison of both structures will provide us more information regarding the
structural alterations during the carcinogenic process.

In brief, after revising the evidence about other mechanisms of gene regulation,
we may hypothesize that the ultimate cause of the distance-dependent gene
clustering is not a single mechanism, but instead, it could be a non-linear
combination of different phenomena. In particular, regarding gene clustering, we
have evaluated for the first time the whole set of gene interactions, and the
loss of long-distance co-expression remains, which is more evident in the most
aggressive subtypes.

Homogeneity/redundancy promotes higher entropy. Systems with redundancy are less
likely to fail to catastrophic events. In other words, it seems there are
mechanisms that give robustness to gene regulation in a control phenotype. It is
still uncertain whether the loss of long range (or gain in short range) gene
co-expression is a consequence of cancer, forcing the system to work in a less
entropic configuration, but it seems that this preference for a less entropic
configuration is common in all cancer subtypes and is consistently progressive
with subtype aggressiveness.

As a summary of findings, we may establish the following:

● We used tools previously implemented in time series analysis in the stock
market and neuroscience settings (49–51) to develop a systematic, data-driven
method for intra-chromosomal gene expression clustering. Using spectral
decomposition and a null model, we were able to determine the number of
co-expressed group of genes to perform k–medoids algorithm calculations and
determine the most accurate clustering configuration. This method allowed us to
have significant results, avoiding to set an a priori threshold for
co-expression values.

● In the adjacent normal phenotype matrices, negative and positive correlations
are significant throughout the entire chromosome. On the other hand, in breast
cancer, negative correlations are observed in the same rank than those from a
null model (see Figure 2); furthermore, the positive ones are only out of the
null model cloud over short distances.

● In cancer, clustering mostly occurs between nearby genes, unlike what happens
in the adjacent normal phenotype matrices. This is a representation of high
co-expression over short distances. This fact coincides and corroborate previous
results on mutual information-based co-expression networks in these and other
types of cancer (10, 12–14).

● The intra-cluster Nearest Neighbor Distance (NND) clearly decays from the
adjacent normal network to those cancerous ones. Additionally, the NND for
breast cancer networks also decays according to the aggressiveness of the
subtype: Luminal A, Luminal B, HER2+ and Basal.

● Analogously to the last point, Kolmogorov-Smirnov (KS) distance between the
Cumulative Distribution Function of the NND in the adjacent normal and each
breast cancer subtype network, increases with the aggressiveness of the subtype,
thus indicating that the larger value of the KS distance, the larger difference
between adjacent normal and breast cancer phenotypes’ networks.

Clustered genes may be subject to further analyses to reveal, for instance,
statistical enrichment of functional categories revealing certain biological
functions, additional patterns of coordinated activity, etc. This in turn may
lead to the generation of hypotheses to be tested via more narrowly targeted
assays and interventions.

A closer look at matrices’ patterns generated by other type of sorting methods
may shed some light on possible mechanisms behind the regulatory changes in
co-expression and perhaps even in the establishment of the tumor phenotypes.
This is, indeed, still ongoing work.

Further steps towards the understanding of co-expression patterns and the
differences in clustering among adjacent normal and cancerous phenotypes may be
also based on the usage of multi-layer approaches (11, 70).

There are remaining questions prompted by this study. For example, while it is
evident that there is a decay in the strength of correlations depending on the
distance in all chromosomes, it is not fully clear what is the origin of the
differences in the slope of the aforementioned decays. Also, the negative
correlations in adjacent normal network are significant, independently of the
position in the chromosome. Is the anti-correlation between genes a possible
mechanism of negative feedback? Is that mechanism disrupted in breast cancer?
Another important question regarding the clustering in cancer network is the
size of the clusters. Is there an “optimal” cluster size for cancer networks? If
so, what is the rationale behind such number?

Finally, the fact that other types of cancer, which have been analyzed in terms
of gene co-expression interactions, such as clear cell renal carcinoma (13),
lung adenocarcinoma, or lung squamous cell carcinoma (12) have been reported to
have the same bias in short-distance interactions, a remaining question is
whether the clustering behavior observed in breast cancer subtype networks is a
conserved phenomenon along other cancer types.

The above mentioned questions, together with the acquired knowledge on cancer
networks, will be eventually answered and that will bring us with complementary
information to have a broader point of view on gene regulation in cancer.


DATA AVAILABILITY STATEMENT

The datasets presented in this study can be found in online repositories. The
names of the repository/repositories and accession number(s) can be found below:
https://github.com/josemaz/gene-matrices.


AUTHOR CONTRIBUTIONS

AG-E and JZ-F performed computational analyses, developed methods and
implemented programming code, performed pre-processing and low-level data
analysis, participated in the writing of the manuscript. EH-L contributed to the
design of the study, co-supervised the project, contributed and supervised the
writing of the manuscript. JE-E conceived and designed the project, performed
calculations, supervised the project, drafted the manuscript. All authors
contributed to the article and approved the submitted version. AG-E and JZ-F
contributed equally as first author.


FUNDING

This work was supported by the Consejo Nacional de Ciencia y Tecnología
[SEP-CONACYT-2016-285544 and FRONTERAS-2017-2115], and the National Institute of
Genomic Medicine, México. Additional support has been granted by the Laboratorio
Nacional de Ciencias de la Complejidad, from the Universidad Nacional Autónoma
de México.


CONFLICT OF INTEREST

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.


PUBLISHER’S NOTE

All claims expressed in this article are solely those of the authors and do not
necessarily represent those of their affiliated organizations, or those of the
publisher, the editors and the reviewers. Any product that may be evaluated in
this article, or claim that may be made by its manufacturer, is not guaranteed
or endorsed by the publisher.


ACKNOWLEDGMENTS

The results shown here are in whole or part based upon data generated by the
TCGA Research Network: https://www.cancer.gov/tcga.


SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at:
https://www.frontiersin.org/articles/10.3389/fonc.2021.726493/full#supplementary-material

Supplementary Figure 1 | Principal component analyses of RNA-seq data clustered
by breast cancer subtypes.

Supplementary Figure 2 | Comparison of the squared components of the four
largest eigenvectors from the correlation matrix b in Figure 1. It can be seen
that there is an overlap between the eigenvectors that does not allow to
separate the components into clusters.

Supplementary Figure 3 | Outliers of positive correlations (red dots) in some
chromosomes of the breast cancer subtype networks. Plots for chromosomes 1, 9,
19 and 17 for the four subtypes. Arrows indicate sets of positive correlations
considered outliers. Notice that the outliers in each plot form almost vertical
lines, indicating that those interactions present approximately the same
distance. Additionally we can see a null model overlapping (blue).

Supplementary Material S1 | Excel file containing cross tables between
subtype-samples and histological variables.

Supplementary Material S2 | Heatmaps of Pearson correlation for each chromosome
in the adjacent normal phenotype. The color code is the same than in Figure 1.

Supplementary Material S3 | Heatmaps of Pearson correlation for each chromosome
in the Luminal A phenotype.

Supplementary Material S4 | Heatmaps of Pearson correlation for each chromosome
in the Luminal B phenotype.

Supplementary Material S5 | Heatmaps of Pearson correlation for each chromosome
in the HER2+ phenotype.

Supplementary Material S6 | Heatmaps of Pearson correlation for each chromosome
in the Basal phenotype.

Supplementary Material S7 to S11 | Pearson distribution scatter plots for normal
adjacent tissue, Basal HER2+, Luminal A and Luminal B, respectively. These plots
show correlations sorted by gene start position for the four cancer phenotypes
and the adjacent normal network per chromosome.

Supplementary Material S12 | Kolmogorov-Smirnov significance tests for all
chromosomes in the five phenotypes. As in Figure 3, the figures represent the KS
statistics of 50 equal-area (same number of data-points) intervals for each
chromosome. In the figures, the phenotypes are represented by different colors.
The associated p-value for the complete set of correlations are depicted in the
upper right part of the figures.

Supplementary Material S13 | Piece-wise permutation p-values of the KS
statistics, calculated for all bins obtained in Supplementary Material S8, in
every chromosomal region for each phenotype.

Supplementary Material S14 | Clusters by chromosome for the five phenotypes,
obtained by eigenvalue decomposition and k-medoids method. The figures are
depicted as in the manuscript. Additionally, this material contains files for
clusters including the name of the gene, the cluster that the gene belong to,
the assignment cost function value, the chromosome location of the gene, and the
gene start position of said gene.

Supplementary Material S15 | Cumulative distribution for the four cancer
phenotypes and the adjacent normal network per chromosome. Color code coincides
with clusters in Supplementary Material S7. For clarity, only the top-ten
clusters were colored.


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Keywords: eigenvalue decomposition, gene co-expression clustering, loss of
long-distance co-expression, co-expression matrices, breast cancer molecular
subtypes

Citation: González-Espinoza A, Zamora-Fuentes J, Hernández-Lemus E and
Espinal-Enríquez J (2021) Gene Co-Expression in Breast Cancer: A Matter of
Distance. Front. Oncol. 11:726493. doi: 10.3389/fonc.2021.726493

Received: 17 June 2021; Accepted: 26 October 2021;
Published: 17 November 2021.

Edited by:

Maria Rodriguez Martinez, IBM Research—Zurich, Switzerland

Reviewed by:

Sheng Liu, Indiana University, United States
Noemi Eiro, Jove Hospital Foundation, Spain
Kimberly Glass, Brigham and Women’s Hospital and Harvard Medical School, United
States

Copyright © 2021 González-Espinoza, Zamora-Fuentes, Hernández-Lemus and
Espinal-Enríquez. This is an open-access article distributed under the terms of
the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) and
the copyright owner(s) are credited and that the original publication in this
journal is cited, in accordance with accepted academic practice. No use,
distribution or reproduction is permitted which does not comply with these
terms.

*Correspondence: Jesús Espinal-Enríquez, jespinal@inmegen.gob.mx



Disclaimer: All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated organizations, or
those of the publisher, the editors and the reviewers. Any product that may be
evaluated in this article or claim that may be made by its manufacturer is not
guaranteed or endorsed by the publisher.



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