Abstract
Cancer research aims to identify genes that cause or control disease
progression. Although a wide range of gene sets have been published,
they are usually in poor agreement with one another. Furthermore,
recent findings from a gene-expression cohort of different cancer
types, known as positive random bias, showed that sets of genes chosen
randomly are significantly associated with survival time much higher
than expected. In this study, we propose a method based on Brouwer’s
fixed-point theorem that employs significantly survival-associated
random gene sets and reveals a small fixed-point gene set for cancers
with a positive random bias property. These sets significantly
correspond to cancer-related pathways with biological relevance for the
progression and metastasis of the cancer types they represent. Our
findings show that our proposed significant gene sets are biologically
related to each cancer type available in the cancer genome atlas with
the positive random bias property, and by using these sets, positive
random bias is significantly more reduced in comparison with
state-of-the-art methods in this field. The random bias property is
removed in 8 of these 17 cancer types, and the number of random sets of
genes associated with survival time is significantly reduced in the
remaining 9 cancers.
Subject terms: Computational biology and bioinformatics, Systems
biology, Biomarkers, Diseases
Introduction
According to the American Cancer Society, cancer is the second most
common cause of death in the US^[28]1. In addition, cancers are
heterogeneous diseases, with comparable diagnoses and identical
treatment regimens resulting in vastly different outcomes for patients.
On the other hand, early cancer diagnosis and prognosis have
substantial effects on patients’ therapeutic targets^[29]2. This has
prompted researchers to seek out factors that can aid in predicting the
course of cancer disease. The findings of numerous studies that relied
solely on clinical characteristics such as lymph node status and
histological grade to classify clinical outcomes demonstrated that
these characteristics were insufficient. This has led to the
development of studies considering genomic data (e.g., gene expression)
alongside clinical features. In general, the goal of such studies was
to select a preferably small number of genes, known as the signature,
and to utilize them in predicting a patient’s survival outcome using
gene expression profiles^[30]3.
Nonetheless, detecting a robust gene set across various datasets that
can accurately predict a patient’s survival outcome has become a key
challenge in cancer research. In the last two decades, numerous
articles have been published on finding survival-relative genes in
various cancer types, each proposing a gene set that was highly
associated with cancer progression and metastasis^[31]4–[32]10.
Nevertheless, there was little overlap between the resulting gene sets
from studies with different cohorts but similar analytical
approaches^[33]11. Therefore, the lack of similarities between the
reported gene sets in these studies indicates that the results depend
on the cohorts being studied. As a result, identifying a robust gene
set across multiple datasets that accurately predicts a patient’s
outcome has become a formidable challenge in cancer research. In this
regard, considering the cancer patient’s survival time is one of the
most critical aspects of finding such a gene set^[34]11.
In 2012, Venet et al.^[35]12 argued this point and conducted a study to
estimate the association between randomly selected gene sets and breast
cancer patient survival time in a Netherlands Cancer Institute (NKI)
cohort. As one might expect, using the expression of random genes to
divide samples into two distinct groups results in groups that are not
significantly different in terms of survival time, and samples are
assigned to each group randomly. In other words, the p-values obtained
from statistical tests comparing survival curves of the groups
generated by random gene sets must be distributed normally, with only
5% of the p-values falling below 0.05^[36]13. By contrast, Venet’s
analysis revealed that in the case of breast cancer, groups generated
by many of the random gene sets showed a statistically significant
difference in survival time. That is to say, these random gene sets
were significantly associated with the patient’s survival time.
Additionally, in some cases, the random gene sets were more
significantly associated with survival time than some of the published
signatures^[37]12. These findings suggest that many of the signatures
identified through breast cancer gene expression analysis may not be
causal to cancer progression, despite being significantly associated
with survival time^[38]13. Venet et al. justified this issue by
pointing to the operation of the proliferation signature, which
considerably impacts a substantial portion of the genome. They
suggested that most random gene sets contain some genes from the
proliferation signature and, thus, are associated with the
proliferation signature and, indirectly, survival. They defined the
meta-PCNA signature to determine the proliferation rate and introduced
a method to remove this signature’s impact on the expression data. They
concluded that removing the effects of meta-PCNA genes on the
expression of the genes in the NKI breast cancer dataset cohort was a
perfect way to reduce the association between random genes and
survival.
In 2018, Shimoni coined the term “random bias” to describe this concept
in cancer^[39]13. Random bias is an unexpected situation in which more
(less) than 5% of random gene sets are associated with some clinical
attribute, such as survival time, in a statistically significant way.
Shimoni examined The cancer genome atlas (TCGA) data for 34 different
cancer types to see if there was a significant association between
random gene sets and survival time. According to his analysis, random
bias could be found in a wide variety of cancer types. Shimoni’s
findings revealed that 17 out of the 34 datasets exhibited positive
random bias, indicating that more than 5% of randomly selected gene
sets in these cancers are significantly associated with survival time.
Ten of these cancer types did not exhibit random bias, while seven of
the datasets exhibited negative random bias, cancers with less than 5%
significant survival-associated random (SSAR) gene sets. Shimoni
utilized Venet’s approach to eliminating the confounder effect of the
proliferation signature from TCGA expression data to reduce the effect
of random bias in all types of cancer. His analysis concluded that
Venet’s methods were ineffective in removing random bias in most cancer
types and impractical in the TCGA breast cancer cohort. To solve this
problem, Shimoni proposed that dividing samples into small subgroups
using an unsupervised clustering method could decrease the proportion
of SSAR gene sets in a wide range of cancer types. Shimoni’s results
showed that out of the 106 clusters generated for all cancer types that
had exhibited both positive and negative random bias, in only 65 of
these clusters, the property of random bias was eliminated. Despite the
fact that random bias was not eliminated in 41 of 106 cases, he
contends that clustering can effectively eliminate random bias in
several TCGA cancer types. Despite producing some promising and
valuable insights, the existing research has produced contradictory
results, is still limited in scope, and faces several critical
theoretical and analytical challenges.
Previous studies have shown that significant survival-associated random
gene sets can provide valuable insights into the biology of breast
cancer and aid in identifying biologically cancer-related genes^[40]14.
Notably, since random bias can be observed in many cancer types, it is
possible that SSAR gene sets may also provide informative results for
most cancer types. Building upon these findings, we assert that each
cancer type has a fixed-point gene set that is biologically associated
with cancer survival time which can be identified by SSAR gene sets.
Additionally, these fixed-point gene sets are responsible for the
observed random bias, and by removing their effects from expression
data, it is possible to decrease the proportion of significant
survival-associated random gene sets. To identify these fixed-point
gene sets, we introduce an iterative novel approach for detecting gene
sets that are not only statistically significant but also biologically
relevant for cancer research and clinical practice. Specifically, we
aim to identify fixed-point gene sets for each TCGA cancer type that
exhibit positive random bias. By applying this approach, we aim to
eliminate positive random bias and reduce the proportion of SSAR gene
sets in a large number of cancer studies. Moreover, we demonstrate that
the identified gene sets are highly biologically significant and can be
considered as signatures for their associated cancer type. This
suggests that the proposed approach can provide valuable insights into
the underlying biology of cancer and improve the accuracy and
reliability of survival analyses in various cancer types. Overall, our
approach provides a systematic and rigorous method for detecting
biologically relevant gene sets associated with cancer survival time
and can have important implications for cancer research and clinical
practice.
Materials and methods
Dataset
Loi et al.^[41]15 collected microarray expression data of 17,585 genes
from 380 individuals with primary breast tumors. The Rdata file was
downloaded from NCBI’s Gene Expression Omnibus (GEO) with accession
number [42]GSE6532
([43]https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=gse6532).
Netherlands Cancer Institute (NKI) cohort
The NKI, also known as the van de Vijver et al. data set^[44]5, was
provided in Venet’s paper^[45]12. This dataset contains microarray
expression data of 13,108 genes for 295 breast cancer patients in
stages I or II and their clinical data.
The Cancer Genome Atlas (TCGA)
The expression data of TCGA cancer types (17 cases) that exhibit
positive random bias (based on Shimoni’s finding) were downloaded from
the [46]https://portal.gdc.cancer.gov site. We looked at level 3 data
normalized using RNA-Seq by Expectation-Maximization (RSEM) method
based on Shimoni’s approach; each dataset contains RNAseq expression
datasets for cancer patients and their survival time and clinical data.
We used standard TCGA study abbreviations for the cancer type names (as
defined in
[47]https://gdc.cancer.gov/resources-tcga-users/tcga-code-tables/tcga-s
tudy-abbreviations).
Method
In this section, we outline our novel approach for identifying the
fixed-point gene set for each cancer type that exhibits the positive
random bias property. Our method is based on an iterative algorithm to
systematically and efficiently identify the gene set of interest. We
explain the steps involved in our approach and provide a detailed
description of how our method works to detect biologically relevant
gene sets associated with cancer survival time.
Fixed-point gene set identifier method (FPGI)
1. Initialize
[MATH: X0=∅ :MATH]
,
[MATH: j=1 :MATH]
.
2. Randomly select a set of genes,
[MATH: Gj :MATH]
, with size m from all genes (Fig. [48]1A).
3. Use principal component analysis (PCA) on the gene expression data
matrix of
[MATH: Gj :MATH]
to divide samples into two equally sized groups (A and B) based on
the median of the first principal component. Test the null
hypothesis that there is no difference in survival time between
these two groups using the log-rank test. If the p-value is less
than 0.05, proceed to the next step. Otherwise, go back to step 2
and increase j by one and choose another random set (
[MATH:
Gj+1
:MATH]
) (Fig. [49]1B).
1. Set
[MATH:
DEG=G
j :MATH]
. Use the Significant Analysis of Microarrays (SAM) method to
detect differentially expressed genes between groups A and B,
and consider the first m genes (most significant genes) as
[MATH:
DEG′ :MATH]
(Fig. [50]1C).
2. Compare the sets
[MATH:
DEG′ :MATH]
and DEG. If they were not the same, set
[MATH:
Gj=DEG′ :MATH]
and go back to step 3.
3. If
[MATH:
DEG′ :MATH]
and DEG were the same, set
[MATH:
Xj=Xj-1∪DEG :MATH]
and go to step 4.
4. Increase j by one and go back to step 2.
5. Continue the whole process for
[MATH:
j=1,…,<
mn>6000 :MATH]
and identify
[MATH:
ZC=X6000 :MATH]
as the fixed-point gene set of cancer type C that exhibits positive
random bias property.
Figure 1.
[51]Figure 1
[52]Open in a new tab
The figure depicts the FPGI method for identifying a fixed-point gene
set in a given cancer type. The method starts with
[MATH: j=0 :MATH]
, and an empty set
[MATH: X0 :MATH]
and proceeds iteratively for
[MATH:
j=1,…,6
000 :MATH]
. At each iteration, a set of genes (
[MATH: Gj :MATH]
) is randomly selected from all genes expression data (GE all genes)
and used to construct an gene expression data matrix (GE random genes)
(part A). The samples are then divided into two groups A and B using
principal component analysis (PCA), and their survival time is compared
using log-rank test and p-value (part B). If the p-value is less than
0.05, set
[MATH:
DEG=Gj :MATH]
and the first 50 differentially expressed genes (
[MATH:
DEG′
:MATH]
) between A and B are identified using the Significance Analysis of
Microarrays (SAM) method (part C). If DEG is the same as
[MATH:
DEG′
:MATH]
, it is added to the fixed-point set
[MATH: Xj :MATH]
, otherwise, the process is repeated with
[MATH:
DEG′
:MATH]
as
[MATH: Gj :MATH]
until convergence. In this method, j represents the iteration number.
The final result,
[MATH: X6000 :MATH]
, is considered as the fixed-point set of the cancer type C.
Why FPGI method converges?
This iterative method tries to identify gene sets associated with
positive random bias in cancer samples. This method combines Principal
Component Analysis (PCA) to divide samples into two groups with
Significant Analysis of Microarray (SAM) to identify differentially
expressed genes between the two groups of samples resulting from PCA.
This iterative process is repeated until a fixed set of genes is
obtained. Utilizing the SAM method at each iteration ensures the
convergence of our method, which helps to identify gene sets that are
statistically significant and likely biologically relevant. In fact,
the application of the SAM method during each iteration is the primary
contributor to the method’s convergence. SAM is designed to identify
gene sets with statistically significant differences in expression
between two sample groups. Using PCA to refine the search for relevant
gene sets is the second critical factor contributing to the convergence
of the FPGI method. PCA is an efficient method for identifying sample
subpopulations with potentially distinct gene expression profiles. By
dividing the samples into two groups based on the identified gene set
at each iteration, FPGI can identify subpopulations of samples with
comparable gene expression profiles and narrow our search to the most
relevant gene sets. In addition, the stopping criterion of this method,
which requires that the gene set identified in each iteration be
identical to the gene set identified in a previous iteration, ensures
that the method does not continue to iterate forever.
In addition, the stopping criterion of this method, which requires that
the gene set identified in each iteration be identical to the gene set
identified in a previous iteration, ensures that the method does not
continue to iterate forever. To investigate this, we tested our
algorithm on different gene set sizes, including 5, 50, 100, and 200
genes. Our results show that for all scenarios, the algorithm converges
quickly (with a maximum of 20 iterations).
The combination of SAM, and PCA, as well as identifying statistically
significant gene sets at each iteration and refining the search based
on the identified gene set, and the stopping criterion ensures that
FPGI method converges on a fixed set of genes, that are biologically
significant, making it a reliable and robust technique for identifying
relevant gene sets in cancer samples.
Scoring function
Our scoring function, denoted as w, maps from the fixed-point gene set,
[MATH: ZC :MATH]
to the set of natural numbers,
[MATH: N :MATH]
. Specifically, given a gene g, we define its score w(g) as the number
of times g appears in the gene set
[MATH: Xj :MATH]
, where
[MATH: Xj :MATH]
is computed in step 3 part c for
[MATH:
j=1,…,6
000 :MATH]
. Therefore, w(g) provides a measure of the significance of each gene
in
[MATH: ZC :MATH]
by quantifying its frequency of occurrence across iterations.
In the result section, we will discuss how the fixed-point set,
[MATH: ZC :MATH]
is associated with survival time and plays a key role in the phenomenon
of positive random bias for the vast majority of cancer types
exhibiting this property. Despite the fact that the results presented
in this section pertain solely to BRCA, the same conclusions hold true
for the other 16 types of cancer analyzed.
Results
Frequency of genes in fixed-point set
To ensure comprehensive coverage of the search space and the inclusion
of all available genes in the dataset, we generated the union of 6,000
random gene sets, each containing 50 genes. This union resulted in a
multi set of 300,000 genes. For the BRCA dataset, the union set
contained all of the 18,275 genes of the dataset at least once, and
some genes were repeated up to 34 times. Thus, our iterative method,
which starts from a random gene set and repeats 6000 times, covers all
genes in the dataset, giving each gene a chance to be chosen in a
random gene set.
On the other hand, the fixed-point set of BRCA (
[MATH: ZBRCA :MATH]
) consists of only 295 genes, with a maximum and minimum frequency of
397 and 52 respectively. Compared to the original dataset, this is a
very small set of genes. These results demonstrate that our method
attempts to cover the entire search space and starts from all available
genes in the dataset for each cancer type. Eventually, it settles on a
small subset of genes that is a subset of the corresponding cancer type
genes.
The scoring values for the 50 top genes in
[MATH: ZBRCA :MATH]
are shown in Fig. [53]2. We choose this number of genes for the figure
because we mainly report our results using a random gene set size of 50
throughout most of the paper, even though we analyzed our method using
random gene sets of size 50, 100, and 200. It is noteworthy that the
scores of the top 50 genes in
[MATH: ZBRCA :MATH]
ranged from 346 to 397. The frequencies of these genes in the random
gene sets and in
[MATH: ZBRCA :MATH]
are plotted in blue and orange bars, respectively. Similar plots for
other cancer types are available in Supplementary Information File
[54]1, and they follow the same pattern.
Figure 2.
[55]Figure 2
[56]Open in a new tab
Frequency of genes in all random gene sets versus scores of the
fixed-point gene set. The orange bars on the y-axis represent the 50
top scoring genes in the breast. In random gene sets, the scoring value
of these genes is represented by blue bars.
Overall, we observe that the
[MATH: ZC :MATH]
set is significantly smaller than the original dataset. This suggests
that the genes in the
[MATH: ZC :MATH]
set may be important for cancer progression. To investigate this claim,
we analyze the biological relevance of
[MATH: ZC :MATH]
to cancer C in the following sections.
Protein-protein interaction (PPI) network and pathway enrichment analysis
In the first step of our analysis, in order to determine functional
interactions between proteins coding genes of the resulted
[MATH: ZC :MATH]
, we used the Search Tool for Retrieval of Interacting Genes (STRING)
database [57]https://string-db.org^[58]16. The PPI network was
constructed using active interaction sources such as text mining,
experiments, databases, neighborhood, gene fusion, co-occurrence, and
co-expression, and a species was restricted to “Homo sapiens”. The
nodes in the network represented the proteins, while the edges
reflected the interaction. In STRING, each protein-protein interaction
is annotated with one or more ‘scores’, these scores are indicators of
confidence. Each score is assigned a confidence level between 0 and 1,
with 1 representing the highest level of confidence. To obtain more
reliable findings, we used a score of 0.9 for the confidence of
interactions. The
[MATH: ZBRCA :MATH]
gene-based network consisted of 113 non-isolated nodes and 810 edges.
The constructed network has a PPI enrichment p-value less than 1.0e−16,
indicating that interactions between genes were not random. This
suggested that the
[MATH: ZBRCA :MATH]
genes interacted more frequently than would be predicted for a random
collection of proteins with the same size and degree distribution (in
this case expected number of edges is 84). This enrichment indicated
that the proteins as a group are biologically related. As it is
illustrated in Fig. [59]3, the network, which was reduced to none
isolated nodes contained some distinct, dense modules. Similar results
were observed for other cancer types, which are available in
Supplementary Information File [60]2. These results indicated that the
genes in the
[MATH: ZC :MATH]
sets were highly associated with one another.
Figure 3.
[61]Figure 3
[62]Open in a new tab
STRING protein-protein interaction analysis of the fixed-point gene set
of the BRCA dataset. The network contained 113 nodes and 810 edges (vs.
84 expected edges); enrichment p-value less than 1.0e
[MATH: - :MATH]
16. Figure were constructed using the STRING database (version 11.5;
[63]https://string-db.org/).
Accordingly, in the second step, Kyoto Encyclopedia of Genes and
Genomes (KEGG) pathway enrichment analyses were conducted using The
Database for Annotation, Visualization and Integrated Discovery (DAVID)
([64]https://david.ncifcrf.gov) to find corresponding significant
pathways of
[MATH: ZC :MATH]
for each of the 17 cancer types^[65]17,[66]18. Supplementary
Information Fig. [67]1 shows the significant pathways common in at
least two types of cancer. In this figure, the names of the pathways
are listed in the first column, and the pathways associated with each
cancer type are depicted in the second column using different colors.
As presented in Supplementary Information Table [68]1, most of the
significantly enriched pathways of
[MATH: ZC :MATH]
were highly associated with cancer C.
Association of fixed-point set with disease
The Genetic Association Database (GAD) tool on the David Functional
Annotation server ([69]https://david.ncifcrf.gov) was utilized to
investigate the association between
[MATH: ZC :MATH]
genes and disease. GAD is a database of published genetic association
studies that enable the investigation of complex common human genetic
diseases^[70]17,[71]18. Table [72]1 demonstrates the enriched disease
and the disease class for each of the 17 cancer types. The table
suggested that the genes in
[MATH: ZC :MATH]
were associated with cancer C. For instance, in the case of BRCA, the
top-level disease class and disease assigned by GAD were cancer and
breast cancer with p-values of 2.8e
[MATH: - :MATH]
4 and 6.5e
[MATH: - :MATH]
8, respectively. As noted by Ansar et al.^[73]14, these findings
indicated how our method can detect meaningful information in SSAR gene
sets.
Table 1.
Enriched disease and disease class achieved from fixed-point sets by
Genetic Association Disease (GAD).
Dataset GAD disease class Class p-value GAD disease p-value
ACC Cancer 2.2e
[MATH: - :MATH]
3 Plasma HDL cholesterol (HDL-C) levels 9.9e
[MATH: - :MATH]
6
BLCA Cancer 9.2e
[MATH: - :MATH]
6 Urinary bladder neoplasms 1.8e
[MATH: - :MATH]
2
BRCA Cancer 2.8e
[MATH: - :MATH]
4 Breast cancer 6.5e
[MATH: - :MATH]
8
GBMLGG Cancer 2.8e
[MATH: - :MATH]
4 Schizophrenia 2.0e
[MATH: - :MATH]
6
HNSC cardiovascular 1.6e
[MATH: - :MATH]
5 Cardiomyopathy, Dilated|DCM—Dilated cardiomyopathy 3.7e
[MATH: - :MATH]
6
KIPAN Cancer 7.5e
[MATH: - :MATH]
4 Type 2 Diabetes| edema | rosiglitazone 7.5e
[MATH: - :MATH]
5
KIRC Cancer 2.5e
[MATH: - :MATH]
6 Chronic renal failure|Kidney failure, Chronic 1.9e
[MATH: - :MATH]
3
KIRP Cancer 3.1e
[MATH: - :MATH]
5 Type 2 Diabetes| edema | rosiglitazone 9.6e
[MATH: - :MATH]
6
LGG Pharmacogenomic 2.3e
[MATH: - :MATH]
5 Several psychiatric disorders 2.0e
[MATH: - :MATH]
6
LIHC Cancer 9.0e
[MATH: - :MATH]
5 Liver cancer 6.1e
[MATH: - :MATH]
2
LUAD Cancer 2.4e
[MATH: - :MATH]
17 Lung cancer 2.0e
[MATH: - :MATH]
6
LUSC Cancer 9.3e
[MATH: - :MATH]
4 Lung Diseases|Resp distress syndrome neonatal 2.1e
[MATH: - :MATH]
6
MESO Cancer 1.1e
[MATH: - :MATH]
3 Lung cancer 2.5e
[MATH: - :MATH]
2
PAAD Cancer 1.5e
[MATH: - :MATH]
2 Type 2 Diabetes| edema | rosiglitazone 8.2e
[MATH: - :MATH]
4
THYM Immune 1.6e
[MATH: - :MATH]
2 Pulmonary disease, Mycobacterium malmoense 1.5e
[MATH: - :MATH]
4
UCEC Cancer 2.0e
[MATH: - :MATH]
4 Dermatitis, Atopic 1.3e
[MATH: - :MATH]
6
UVM Cancer 1.3e
[MATH: - :MATH]
2 Uveitis 3.8e
[MATH: - :MATH]
2
[74]Open in a new tab
Association of top scoring genes of fixed-point set with cancer C
Through pathway enrichment, PPI network analysis, and disease class
association, the biological significance of
[MATH: ZC :MATH]
with respect to its corresponding cancer type was investigated in the
previous sections, and it was determined that the obtained fixed-point
gene sets were significantly associated with cancer progression and
metastasis. Although, the most significant advantage of our method was
that the high-scoring genes of
[MATH: ZC :MATH]
in many studies had been shown as cancer driver genes. In the ACC, CD68
gene, the highest-scoring gene in
[MATH: ZACC :MATH]
, has been identified as a prognostic biomarker for adrenocortical
carcinoma^[75]19. As another example, the TBX2 and TBX3 genes, with
scores of 1276 and 1226 in the fixed-point gene set of BLCA, were
excellent markers for predicting progression to muscle-invasive bladder
cancer in patients with primary pTaG1/2 bladder cancer^[76]20.
In^[77]21, it has been proposed that the C6orf97 gene, the highest
scoring gene of
[MATH: ZBRCA :MATH]
, might play important roles not only in carcinogenesis but also in the
progression of breast cancer patients toward a more aggressive
phenotype. In 2019, Yeng et al.^[78]22 had indicated that ACTA1, a gene
from
[MATH: ZHNSC :MATH]
with a score of 245, was a biomarker of head and neck squamous cell
carcinoma. As another instance, silencing of ANK2 with a score of 358
in
[MATH: ZPAAD :MATH]
decreased the proliferation of the pancreatic tumor cells and reduced
their tumorigenicity in vitro and in vivo^[79]23. The highest-scoring
genes across all
[MATH: ZC :MATH]
sets are depicted in Supplementary information Fig. [80]2 and,
Published papers that have investigated their association with cancer
are available in Supplementary information Table [81]2.
Random bias correction
The random bias phenomenon, as described by Venet et al. and Shimoni,
suggested that many of the signatures identified in numerous analyses
of cancer types might not be causal of cancer progression, despite
their significant association with survival time. Consequently, random
bias is a confounding property that must not be ignored^[82]12,[83]13.
As proposed by Venet et al. random bias was caused by the activity of
proliferation genes (meta-PCNA gene signature) in data that had a
substantial impact on the expression data, and the activity of this
signature significantly influences each random set selected from the
data. They hypothesized that by removing the effect of meta-PCNA genes
from expression, the random bias in the NKI breast cancer dataset could
be eliminated^[84]12. However, Shimoni demonstrated that removing the
impact of meta-PCNA genes could not effectively reduce the proportion
of SSAR gene sets in TCGA cancer types. Venet’s strategy might depend
on the platform or data^[85]13. In this paper, we claimed that for each
cancer type, we required a specific set of genes that removing its
impact on the expression data could reduce the proportion of SSAR gene
sets. To investigate this claim, we used the fixed-point gene set (
[MATH: ZC :MATH]
) of each cancer type and demonstrated that removing the influence of
these genes from expression data could dramatically reduce the
proportion of SSAR gene sets in the vast majority of cancer types. To
accomplish this, we selected 10% of the highest scoring genes of
[MATH: ZC :MATH]
and then removed their impact from the expression data similar to
Venet’s approach^[86]12. The result of this analysis is available in
Table [87]2, where rows denote the cancer type and the proportion of
significant p-value in percentage (SSAR%), the proportion of
significant random gene set after removing the effect of meta-PCNA in
percentage (PCNA-SSAR%), and the proportion of significant SSAR gene
set after removing the effect of corresponding
[MATH: ZC :MATH]
(
[MATH: ZC :MATH]
-SSAR) are collected in first, second and third columns, respectively.
As shown in Table [88]2, the proportion of significant (positive random
bias) in 14 out of 17 cancer types was significantly more reduced by
using the selected genes from
[MATH: ZC :MATH]
rather than meta-PCNA.
Table 2.
The proportion of significant survival associated random gene sets
after removing the fixed-point set and meta-PCNA signature.
Dataset SSAR% PCNA-SSAR%
[MATH: ZC :MATH]
-SSAR%
ACC 71 40 5
BLCA 50 44 7
BRCA 21 18 5
GBMLGG 99 85 25
HNSC 26 27 20
KIPAN 64 26 27
KIRC 82 68 26
KIRP 57 17 5
LGG 80 64 19
LIHC 32 5 7
LUAD 49 18 11
LUSC 14 10 6
MESO 53 20 15
PAAD 45 7 13
THYM 16 18 7
UCEC 58 41 27
UVM 51 45 7
[89]Open in a new tab
“SSAR%” is the proportion of significant survival-associated random
gene sets, “PCNA-SSAR %” is the proportion of significant
survival-associated random gene sets after removing the effect of
meta-PCNA signature from their expression data, and “
[MATH: ZC :MATH]
-SSAR%” is the proportion of significant survival associated random
gene set after removing the effect of
[MATH: ZC :MATH]
gene set from expression data.
From another point of view, Shimoni used the PhenoClust, an
unsupervised clustering method, to reduce the effect of random bias in
TCGA cancer types^[90]13,[91]24. In Shimoni’s approach, the samples of
each cancer type have been divided into sub-clusters, the association
between each random gene set and the survival time of each cluster’s
samples have been determined, and the proportion of SSAR gene sets was
calculated for each cluster. Samples of all 17 cancer types with
positive random bias property have been divided into 92
clusters^[92]13. In Fig. [93]4 proportion of SSAR gene sets is
represented with colored dots and the small horizontal grey lines show
the proportion of SSAR gene sets after removing the effect of
corresponding
[MATH: ZC :MATH]
. Our evaluation of the performance of the proposed method was based on
the assumption that a method is considered perfect for removing the
random bias property if the results of the method cause only 0.05 of
the random sets to remain significant. We compared the performance of
our proposed method, FPGI, with that of Shimoni’s clustering method
using the equation presented in the Supplementary Information File
[94]3, where
[MATH:
SSARCL(i) :MATH]
denotes the proportion of significant random gene sets of cluster i of
cancer C, and
[MATH: NC :MATH]
represents the number of generated clusters for the corresponding
cancer type. We evaluated the distance between the proportion of SSAR
gene sets after excluding the effect of 10% of the highest scoring
genes of
[MATH: ZC :MATH]
from the expression data, denoted by
[MATH: DZC
:MATH]
, to demonstrate the effectiveness of our method. The results of this
analysis are presented in Supplementary Information File [95]3 table.
This table indicates the cancer types with positive random bias
property in the first column,
[MATH: ADC
:MATH]
for each cancer type in the second column, and
[MATH: DZC
:MATH]
in the third column. Our results demonstrate that, in 9 out of 17
cancer types,
[MATH: DZC
:MATH]
is less than
[MATH: ADC
:MATH]
, which indicates that our proposed method outperforms Shimoni’s
clustering method. In 5 out of the 8 remaining cancer types the results
of these two methods are comparable^[96]12,[97]13. While it is true
that our method shows better results compared to a specific cluster and
not all clusters, we believe that this comparison still provides
valuable insight into the performance of our proposed method. Moreover,
our proposed method provides an alternative explanation to the same
problem and also provides a significant set of genes to continue
exploration.
Figure 4.
Figure 4
[98]Open in a new tab
Each horizontal line represents a TCGA cancer type with positive random
bias property. Each dot along the x-axis represents the proportion of
significant survival-associated random gene set in each cluster. The
short vertical gray line illustrates the proportion of significant
survival-associated random gene set after removing the effect of
[MATH: ZC :MATH]
genes from expression data.
Reproducibility
In cancer research, identifying a reliable gene set independent of
datasets and can accurately predict a patient’s survival outcome has
become a major obstacle. Numerous articles on discovering
survival-relative genes in various cancer types have been published
over the recent decades, each proposing a gene set, with the authors
asserting that the purpose gene set was significantly associated with
cancer progression and metastasis. However, there was little overlap
between the gene sets resulting from studies with different cohorts but
similar analytic methods. In this paper, we introduced a set of
significant survival-relative genes and indicated that we could reduce
the proportion of SSAR gene sets by employing them. We demonstrated
that these finding sets were nearly robust across different cancer
cohorts. For this reason, we evaluated our results using various breast
cancer cohorts and data sets. Regarding this, we considered three
distinct breast cancer cohorts (NKI, TCGA, and LOI)^[99]5,[100]15. For
the NKI, TCGA, and LOI cohorts, the resulting
[MATH: ZC :MATH]
sets contained 364, 295, and 426 genes, respectively. Each pair of
these three gene sets has more than 30% of their genes in common, and
37 genes are in the intersection of all
[MATH: ZC :MATH]
sets that contained critical genes for breast cancer like AURKA and
AURKB^[101]25,[102]26. Also, the PPI network based on these 37 genes
was very dense, and the corresponding pathways of these genes were
significantly associated with breast cancer (see Supplementary
Information Fig. [103]3). It could be concluded that our proposed
method for identifying survival-associated genes demonstrated
significant overlap across different cohorts.
Discussion
It has previously been shown that gene expression data from random gene
sets have a significant relation with cancer survival time, which has
been discovered in a variety of cancers. Venet et al.^[104]12 first
discovered this phenomenon in a microarray-measured expression dataset
in breast cancer. As it turns out, this pattern can be seen in nearly
all of the TCGA data matrix’s RNAseq-derived gene expression data.
Venet et al.^[105]12 hypothesized that the phenomenon of random bias
observed in gene expression data is caused by the activity of the
proliferation signature, which affects a substantial portion of the
human genome, and by removing the effect of this signature from
expression data the random bias property will be removed. However, As
reported by Shimoni^[106]13, the proliferation signature alone is
insufficient to eliminate this bias in most cancers. While we agree
with the general assumption of Venet et al. we believe that the
specific set of genes that contribute to positive random bias varies
widely between cancer types.
To address this issue, we propose the existence of a fixed-point gene
set for each type of cancer, which exerts a strong influence on a large
number of genes in the genome and is strongly associated with survival
time. This fixed-point gene set can induce survival prediction ability
in randomly selected gene sets derived from expression data. To
identify these gene sets, we developed an innovative and iterative
method .
The iterative nature of our method is a key strength that enables us to
identify gene sets that are not the result of chance or noise, but
instead represent significant differences in gene expression between
groups. This is achieved by repeatedly dividing samples into two groups
based on the differential expression of a gene set, and using this
information to identify a new, refined gene set. The iterative process
enables us to focus on the most biologically relevant genes for a given
cancer type, and exclude genes that may be false positives or
irrelevant to the disease.
By applying this method to multiple cancer types, we can build a more
comprehensive understanding of the underlying molecular mechanisms
driving cancer development and progression. Furthermore, because our
method is based on statistical significance, we can be confident that
the gene sets we identify reflect genuine differences in gene
expression between cancer types. This in turn gives us greater
confidence in the biological relevance of the genes we identify, and
increases the potential for these genes to be used as diagnostic or
therapeutic targets in the future.
In our study, we have analyzed a wide range of cancer types, and in
order to validate the biological relevance of the identified gene set,
we conducted protein-protein interaction (PPI) network and pathway
analyses. The PPI network analysis helped us identify key biological
pathways and processes involved in cancer development and progression
and how the fixed-point set was related to these pathways. We also
compared our gene set with previously published cancer signatures and
confirmed that our identified gene set was highly correlated with the
known cancer pathways.
In addition, we evaluated the association of our fixed-point set with
cancer disease class and related cancer types using the Genetic
Association Database (GAD). The results showed that our gene set was
highly associated with cancer disease class and related cancer types,
providing further evidence of the biological relevance of our
identified genes.
To ensure that our findings were not dataset-specific, we reanalyzed
our method with other independent datasets. The results showed that our
identified fixed-point set of genes was consistently present across
different datasets, further validating our approach and increasing the
confidence in our results.
Overall, our study has identified a set of highly cancer-related genes
using an iterative approach that eliminates false positives and ensures
the biological relevance of the identified genes. The validation of our
gene set through various approaches and its consistency across multiple
cancer types and datasets further supports its potential as a
diagnostic or therapeutic target for cancer treatment.
FPGI method is inspired by Brouwer’s fixed-point theorem, to identify
this fixed-point gene set. Brouwer’s fixed-point theorem states that if
you have a continuous function f that maps a compact, convex set X onto
itself, and function f is a contraction, that it reduces distances
between points in X by a constant between 0 and 1, then there is always
a point Z in X such that
[MATH: f(z)=Z :MATH]
. In other words, X contains a fixed point that does not move as a
result of the function f. In this work, we drew inspiration from this
theorem by considering the set of all subsets of size m from all genes
to be the compact, convex set X, the symmetric difference of the sets
to be the metric on X, and the composition of the SAM and PCA methods
to be the continuous function on X.
To create this method, we first use a technique for detecting a
significant relationship between gene expression data and cancer sample
survival time in order to divide the group into two equal-sized
subgroups with different survival dynamics. Second, we use a method to
find genes whose expression significantly differs between these two
subgroups.
We used PCA to estimate the association between a randomly chosen set
of genes and survival time. Specifically, we calculated the median of
the first principal component, and then based on its values the
patients were divided into two equaled-sized groups (A and B). The use
of PCA as a method to estimate the association between a gene set and
survival time has been previously validated in the literature, and we
chose this approach because based on Venet’s results PCA method has
been shown to reveal stronger outcome associations than other methods
such as, kmeans and hierarchical clustering^[107]27–[108]33. Moreover,
the use of PCA in this method allows for a more efficient and effective
identification of genes that are relevant to patient survival, as it
reduces the number of variables needed to analyze the data and
highlights the most important variables. In summary, PCA is a widely
used statistical technique that we used to estimate the association
between a randomly chosen set of genes and survival time. The use of
the first principal component as a prognostic score has been validated
in the literature and has been shown to reveal stronger outcome
associations than other methods. In addition, there are many well-known
methods to detect differentially expressed genes between two different
groups. Recent studies showed that our method of choice (SAM methods)
is a stronger approach^[109]2.
As mentioned in the result section, in 8 out of 17 cancer types by
removing the effect of fixed-point, positive random bias was removed
and the number of random significant gene sets was reduced to a
significant level of 5% and in other 9 datasets the proportion of SSAR
gene set was dramatically reduced. For instance, in case of GBMLGG it
was reduced from 99 to 26%. In general, as explained in previous
sections, various studies proposed a set of genes that are
significantly associated with cancer progression and metastasis;
however, the fact that many random gene sets may exist with similar
association undermine their identity. We observed that our suggested
sets are truly causal, in the sense that altering their expression or
activity will influence survival where by removing the effect of these
genes’ expression from our dataset, the association of such random set
with survival, is eliminated. Since the choice of association method
and also methods for detecting significant genes is of course crucial
and has a substantial impact on our results, we can examine alternative
methods to enhance our findings in future research.
To assess the robustness of our method, we conducted additional
experiments by running our algorithm on random gene sets of size 100
and 200 for BRCA cancer. We observed that our method consistently
identified nearly the same set of genes as the fixed-point set,
regardless of the size of the set. Specifically, we found a high degree
of overlap between the fixed-point sets obtained for sizes 50, 100, and
200, demonstrating the robustness and consistency of our method to the
choice of random gene set size. These results indicate that our method
is robust and reliable for identifying the fixed-point gene set in
different random gene sets of varying sizes. The details of this
analysis are provided in the Supplementary Information File [110]4. The
file contains a figure and three tables that compare the selection of
random gene sets of size 50, 100, and 200 genes for the fixed-point
analysis. Tables of the Supplementary Information File [111]4 report
the top ten highest scoring genes of
[MATH: ZBRCA :MATH]
identified by random gene sets of sizes 50, 100 , and 200,
respectively. In addition, figure of the Supplementary Information File
[112]4 illustrates the proportion of significant survival associated
random gene sets before and after removing
[MATH: ZC :MATH]
.
Conclusion
Our study introduces a novel method that utilizes significant
survival-associated random gene sets to identify a fixed-point gene set
for cancers with positive random bias. The aim of this method is to
identify a small set of genes specific to each cancer type that
significantly affects survival time, referred to as the fixed-point
gene set. This gene set remains stable across different random gene
sets and serves as a core biological process underlying cancer
progression for each specific cancer type. We expect our algorithm to
converge to a similar fixed-point gene set that consistently affects
survival time in different random samples from the same cancer type.
Our approach combines Principal Component Analysis (PCA) and
Significance Analysis of Microarrays (SAM) methods to reduce noise and
approach the fixed-point gene set in each iteration. The empirical
results on multiple cancer types demonstrate that our method
effectively eliminates the random bias and improves the accuracy of
survival prediction in gene expression data.
Our study also highlights the biological significance of the
[MATH: ZC :MATH]
genes and their association with cancer-related pathways. By using
multiple studies, we show that the highest-scoring
[MATH: ZC :MATH]
genes are strongly associated with the progression and metastasis of
their respective cancer types. Removing the effect of 10% of the
highest-scoring genes on
[MATH: ZC :MATH]
from the expression data drastically reduces the proportion of random
significant survival-associated gene sets and, in some cases,
eliminates the positive random bias phenomenon.
Supplementary Information
[113]Supplementary Figure 1.^ (15.4MB, tiff)
[114]Supplementary Figure 2.^ (7.7MB, tiff)
[115]Supplementary Figure 3.^ (14.6MB, tiff)
[116]Supplementary Information 1.^ (2.5MB, pdf)
[117]Supplementary Information 2.^ (14.3MB, pdf)
[118]Supplementary Information 3.^ (93.2KB, pdf)
[119]Supplementary Information 4.^ (308.9KB, pdf)
[120]Supplementary Table 1.^ (83.6KB, pdf)
[121]Supplementary Table 2.^ (87.1KB, pdf)
Author contributions
M.M analyzed data and wrote the paper. R.A and C.E supervised the
research. All authors read and approved the final manuscrpit.
Data availability
The datasets and codes can be found in the GitHub repository
[122]https://github.com/maryammagy/FPGI.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
Supplementary Information
The online version contains supplementary material available at
10.1038/s41598-023-35588-5.
References