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Computational and Structural Biotechnology Journal
journal homepage: www.elsevier.com/locate/csbj
Research article
Optimized cell type signatures revealed from single-cell data by
combining principal feature analysis, mutual information, and machine
learning
Aylin Caliskan, Deniz Caliskan
1
, Lauritz Rasbach
1
, Weimeng Yu, Thomas Dandekar
⁎
,
Tim Breitenbach
⁎
Department of Bioinformatics, Biocenter, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
article info
Article history:
Received 18 February 2023
Received in revised form 2 June 2023
Accepted 2 June 2023
Available online 5 June 2023
Keywords:
Single cell analysis
Machine learning
Explainability of machine learning
Principal
Feature analysis
Model reduction
Feature selection
abstract
Machine learning techniques are excellent to analyze expression data from single cells. These techniques
impact all fields ranging from cell annotation and clustering to signature identification. The presented
framework evaluates gene selection sets how far they optimally separate defined phenotypes or cell groups.
This innovation overcomes the present limitation to objectively and correctly identify a small gene set of
high information content regarding separating phenotypes for which corresponding code scripts are pro-
vided. The small but meaningful subset of the original genes (or feature space) facilitates human inter-
pretability of the differences of the phenotypes including those found by machine learning results and may
even turn correlations between genes and phenotypes into a causal explanation. For the feature selection
task, the principal feature analysis is utilized which reduces redundant information while selecting genes
that carry the information for separating the phenotypes. In this context, the presented framework shows
explainability of unsupervised learning as it reveals cell-type specific signatures. Apart from a Seurat
preprocessing tool and the PFA script, the pipeline uses mutual information to balance accuracy and size of
the gene set if desired. A validation part to evaluate the gene selection for their information content re-
garding the separation of the phenotypes is provided as well, binary and multiclass classification of 3 or 4
groups are studied. Results from different single-cell data are presented. In each, only about ten out of more
than 30000 genes are identified as carrying the relevant information. The code is provided in a GitHub
repository at https://github.com/AC-PHD/Seurat_PFA_pipeline.
© 2023 The Author(s). Published by Elsevier B.V. on behalf of Research Network of Computational and
Structural Biotechnology. This is an open access article under the CC BY-NC-ND license (http://creative-
commons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Single-cell technology provides methods to measure gene ex-
pression levels in single cells [1,2]. Each measurement provides a
snapshot of the genetic dynamic within a cell. Assuming that the
genetic dynamics of regulation are identical within the cells, a re-
search question is to find the differences in the expression profile of
a genotype responsible for phenotypic variants. One example is stem
cells that differentiate into different cell types to create specific
tissues [3]. To analyze the differences between cell types on a ge-
netic level, we need to identify the genes whose expressions are
different in both cell types and appear characteristic of the differ-
ence of the cell types. Our innovation: As mathematical concept we
introduce new principal feature analysis [4] to identify those genes
which carry the highest information suitable to separate the groups
we want to distinguish. It does not matter how many groups we
want to distinguish nor by which method (or combination of
methods) the different groups were established first. As long as our
mathematical approach has not yet been applied to the data, we can
use it to improve the analysis in identifying the genes with the
highest information content regarding separation of the groups.
Gene selection methods: After explaining the core concept of
our gene selection method, we examine existing methods to better
explain differences between gene groups and then how the results
obtained by these gene selection methods can be improved applying
our method of principle feature analysis to objectively determine
how well each individual gene contributes to separate between
Computational and Structural Biotechnology Journal 21 (2023) 3293–3314
https://doi.org/10.1016/j.csbj.2023.06.002
2001-0370/© 2023 The Author(s). Published by Elsevier B.V. on behalf of Research Network of Computational and Structural Biotechnology. This is an open access article under the
CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
]]]]
]]]]]]
⁎
Corresponding authors.
E-mail addresses: dandekar@biozentrum.uni-wuerzburg.de (T. Dandekar),
tim.breitenbach@mathematik.uni-wuerzburg.de (T. Breitenbach).
1
These authors have equally contributed to this work.
clusters. Some genes are regulated by other genes, like genes in a
pathway, and can thus be seen as a function of those regulating key
genes that act as the pacemaker genes of that function.
Mathematically the pacemaker genes are called “argument
genes”: As an example, if several other genes regulate a gene, we
mathematically write
= …gen f gen gen1 ( 2, 3, )
where the regulating
genes
…gen gen2, 3,
are called arguments of the function
f
that
models the expression of the gene
gen1
. Such function genes are
redundant since if we need
gen1
as an argument for another gene,
we can directly construct the functional relation from
…gen gen2, 3,
.
Keeping these redundant genes would add no further informa-
tion but would rather make the genetic model unnecessary big.
Bigger model sizes can cause issues in drawing conclusions from the
identified genes since too many genes impede an overview of the
relations to be investiaged or combinatorial issues leading to cal-
culation time problems.
Evaluation of gene selection by a principle feature analysis
tool: We illustrate our mathematical concept of principle feature
analysis by implementing it into a handy software tool that provides
a small set of genes. These genes might explain differences between
cell clusters to foster subsequent analyses, e.g., drug target identi-
fication, by clearly directing the view to the relevant genes that carry
the information for the difference.
To identify a small but meaningful set of genes, which make the
difference between two or more classes of cells, we can reduce the
redundancy by removing genes that are – mathematically speaking –
functions of others. Removing these genes facilitates a small set of
genes covering a wide span of relevant information regarding which
cells belong to a specific phenotype and class.
We span the space of information, which describes the differ-
ences between phenotypes, with a basis of stochastically in-
dependent genes (features).
The selected genes mathematically act like a basis known from
linear algebra to span vector spaces. Since the expression of each
gene in the returned set of best-describing genes is formed by genes
independent of each other, each gene carries its own individual and
specific aspect of the difference between the phenotypes.
To solve the task of removing genes that are functions of others,
thus carrying redundant information, and to select only those genes
from the remaining genes necessary for identifying the respective
cell type, we use the principal feature analysis (PFA) [4]. This method
generates a graph where each node represents a gene. Two nodes are
connected by an undirected edge if the corresponding genes take
their expression values not independent of each other. This is tested
with a chi-square test of independence. The key observation is that
functions are linkers between disjunct subgraphs in which the genes
are independent of the genes of the other subgraphs. Genes of each
subgraph act as arguments of the dependent gene (function), de-
termining its expression through a deterministic function, such as
the molecular dynamics of regulation. It is shown that removing
these linkers leaves the arguments. The corresponding “argument
genes” or “pacemaker genes” take their values independent of each
other and carry the information to describe the total system of
regulation of the genes.
Application examples: Once redundancy is removed from the
gene data set, another prerequisite of our method is the existence of
groups in which the single cells are clustered and labeled accord-
ingly. Between these groups or only between a portion of the groups,
e.g., at least between two groups, we would like to obtain a small set
of genes that describes all the relevant differences in their expres-
sion profile. Such labels can be obtained in several ways. One way is
to measure a phenotypic characteristic of cells, like a surface protein.
Another way is to apply clustering techniques to define marker
genes [5–7] that single cells have in common, defining cell types.
In some cases, once these groups/clusters are defined, we would
like to get alternative gene sets to marker genes since, e.g., marker
genes may not be an appropriate drug target. Furthermore, since the
expression of marker genes might also depend on each other to
some degree, our approach complements these techniques in so far
that we focus on providing a minimal and redundancy-free set of
genes being able to distinguish between the clusters.
Another related task might be the annotation of cells [8], where
cells are associated with reference expression patterns or known
marker genes. Furthermore, there are methods using machine
learning techniques to perform cell annotation and enhance clus-
tering of single cells [9–12].
However, our approach does not intend to annotate or to cluster
cells but to contribute to finding genes that make a significant dif-
ference in the expression profile between defined phenotypes/
groups of cells. Thus, the algorithm requires some characteristics
with which we can distinguish cells into clusters. Consequently, our
approach is rather to investigate unknown cases in terms of ex-
pression profile and to direct the focus to relevant differences that
could help define cell types and subsequent reference expression or
marker genes more effectively. Moreover, we would like to con-
tribute to the interpretability of found genes and their annotations,
in particular if gene identification has been done with the help of a
deep neural network [9,10,12].
The aim of our algorithm is to extend or enhance the capabilities
of existing tools in these scenarios.
Overview on analysis flow and techniques: For gene selection
there are a number of different techniques, including gene expres-
sion algorithms, improved clustering, and single-cell sequencing
analysis methods. This allows readers to readily discern how our
concept of evaluation of the gene selection made using PFA differs
from existing gene selection methods and see how our method ex-
tends these existing ones by offering a novel approach to identifying
a concise set of genes that best represents the signature for a certain
group of cells in this analysis step.
To start our analysis, single-cell data obtained via the Single Cell
Portal were processed using the Seurat workflow [13–16] and Dou-
bletFinder [17]. The R package Seurat, which got its name due to its
depiction of single cells resembling a pointillist painting such as
those painted by Georges Seurat [13], integrates in situ RNA patterns
and single-cell data, allowing the mapping of cellular localization
[13]. Additionally, Seurat objects, such as those shared by Emont
et al. (2022) [18] contain various information about the cells, in-
cluding tissue type and cell type. Utilizing this information, we
prepared the datasets employed in our subsequent software pi-
peline.
Next, we implemented several helpful existing methods for gene
selection in our software package. One approach to obtaining dif-
ferences between cell types is to screen for differentially expressed
genes (DEGs). Several RNA-sequencing analysis tools, including
edgeR [19], DESeq2 [20] and Seurat [13–16], are commonly used to
identify DEGs between different groups of samples. The Seurat
package [13–16] offers various test methods for finding differential
gene expression, including the Wilcoxon rank sum test [21], which is
also known as the Mann-Whitney U test [22]. An application ex-
ample of DEG is the design of synthetic locus control regions (sLCR)
[23] to mark cells that show a certain phenotype by expressing a
fluorescent reporter upon a characteristic transcription factor (TF)
profile. In Schmitt et al. (2021) [23], these phenotypes are different
glioblastoma subtypes. After the identification of the DEGs, corre-
sponding TFs and binding sites are identified that regulate the ex-
pression of the characteristic signature genes. Subsequently,
methods are required to select a limited number of short DNA
strands for the sLCR that cover most of the binding sites of the
characteristic TFs (to ensure specificity) since the length of the sLCR
is limited. While in Schmitt et al. (2021) [23], the selection is mainly
done manually, in Breitenbach et al. (2022) [24] a mathematical
framework based on integer optimization is developed to automate
A. Caliskan, D. Caliskan, L. Rasbach et al. Computational and Structural Biotechnology Journal 21 (2023) 3293–3314
3294
the selection process. Our presented selection method might be an
alternative to provide a small number of genes that clearly char-
acterizes the expression differences between phenotypes. Since this
selection has a high distinction power and is small, the number of
corresponding TFs might already be limited as well. Based on the
small number of genes and limited variety, the sLCR could be de-
signed by the approved methods more efficient.
In addition to the DEG methods, there are other methods that try
to find genes with a high discrimination power between cell types
based on their expression level while minimizing the predictive
power the genes have among each other, which reduces redundancy,
in order to minimize the size of the required gene set [25,26]. Cai
et al. (2009) [25] use a greedy solver for the minimize redundancy
(between the selected genes) maximize relevance (with the label/
cell types) algorithm (mRMR) [27]. In Guyon et al. (2002) [25,26], a
selection algorithm based on support vector machines is provided
that cancels genes/features out according to their ability to separate
the labels. Since we use a different technique for gene selection and
our method thus relies on other assumptions than the mentioned
techniques, our method may offer supplementary insights, generate
alternative gene datasets, identify potential drug targets, or facilitate
deeper analysis to better comprehend the differences between the
corresponding clusters.
For the task of gene (or in general feature) selection, respectively,
there are other methods available (details in Li et al. (2017) [28] and
Breitenbach et al. (2022) [4]). An example is the similarity-based
method described by Li and colleagues [28], which does not provide
a redundant-free set of features. Furthermore, a method purely
based on mutual information also provides features that might be
redundant. We remark that redundancy does not always have to
cause a (significantly) bigger set of genes depending on the data set.
On the other hand, the minimize redundancy / maximize relevance
method may suffer from a longer calculation time than the PFA, as
elaborated in Breitenbach et al. (2022) [4] when solved by integer
programming. The sparse learning methods (described in detail in Li
et al. (2017) [28]) use a specific model to select genes. Such an as-
sumption is not necessary for the PFA as it selects genes based on
statistical independence, independent of a concrete model choice.
There are further feature reduction methods in addition to the
PFA. The principal component analysis (PCA) provides the principal
components of the cloud consisting of the data points (expression
pattern of each single cell), which are linear combinations of the
original genes. Another method is the autoencoder which is a neural
network that compresses the original input features into a layer with
fewer neurons (new compressed feature) such that the values of the
neurons of the output layer still equal those of the input layer. By
this procedure, the encoder finds a compression of the input feature
set that can still recover the information. However, both methods
provide transformations of the original dataset, making it challen-
ging to interpret the new (compound) features – in particular, the
non-linear autoencoder case – since the transformation of the ori-
ginal genes to the compound genes is not explicitly given. Rather, the
transformation is only implicitly given by the weights in the neurons
which is not immediately easy to interpret, as discussed in
Breitenbach et al. (2022) [4]. Consequently, it is challenging to tell
which genes are essential in the concrete use case.
Overall, the advantage of the PFA for the gene selection part is
that it is relatively fast and reduces redundancy between the genes,
selecting them from the original dataset, not transforming them into
new compound genes, and opening the door to understandable
models by providing only a small number of genes; this is in line
with Occam’s razor. This principle intends to construct explanations
with the smallest possible set of elements.
We have to keep in mind that all these mentioned methods, in-
cluding our presented method, provide genes whose expression
highly correlates with the difference between the cell types. A
correlation does not necessarily imply causality: a causal relation-
ship is a specific type of correlation. As such, the expression (or lack
thereof) of the identified genes may not necessarily be the cause of
differing cell types; it may simply correlate with the cell type.
Experimentally altering these genes might not significantly influ-
ence the cell type. For this reason, employing multiple selection
methods based on distinct mathematical concepts can generate
different small and meaningful gene sets. This plurality of sets
provides different views of the same problem and thus generates
more chances to find the genes that not only correlate but also cause
the difference between the cell types by, e.g., using a correlating
gene to arrive at genes that also cause the differences which might
be in the set of another method. In addition, all methods rely on
different assumptions on the data according to the used mathema-
tical methods which might sometimes be more or sometimes be less
fulfilled depending on the properties of the data set and thus might
not work. This is a further reason why a broad toolbox of different
gene selection methods is beneficial. Thus, we rather intend to ex-
tend the current toolbox by a new evaluation of the separation
power between clusters achieved and we don’t have the intention to
replace any of these well-working tools for gene selection.
Besides the contribution to the portfolio of methods for ex-
plaining differentiating cell clusters measured by characteristic
markers [1], our pipeline, like others mentioned above for feature
selection, can also be used to analyze the differences in the single-
cell data groups clustered by methods like t-SNE [29] or UMAP
[30,31]. These methods are representatives of unsupervised learning
and project high-dimensional expression data into a plane. The main
goal of these methods is to cluster data points together in the plane
that are also close in the high-dimensional space while keeping the
distance to the other points that are not close in the high-dimen-
sional space. These methods provide an excellent way to visualize
single-cell data and to find single cells with a similar expression
pattern while separating them from single cells with a fundamen-
tally different expression pattern. These clusters might serve as a
data-driven definition of cell types/phenotypes. However, once we
visualize these relations via clusters in the plane, we do not get an
explanation of what exactly made these methods cluster as they
have done. Once the clusters are defined, e.g., by an unsupervised
learning technique like UMAP and the single cells of each cluster are
correspondingly labeled, our pipeline can be used to identify the
genes whose expression patterns describe the difference between
cell clusters.
Consequently, a further contribution of our pipeline is in the area
of explainable artificial intelligence (AI) since we may construct
understandable reasoning for the difference from the identified
genes. This is specifically within the scope of our framework since
the pipeline is optimized for providing a small gene set in the ori-
ginal feature (gene) space that is meaningful for explaining the
difference. The reasoning for the explainability is that based on the
expression level of the identified genes, we show the existence of a
function/model that predicts to which cluster a single cell belongs.
Thus, these genes contain relevant information and thus an ex-
planation for the clustering that the unsupervised learning con-
structed could be reasoned with these genes. Since the set is
meaningful, it contains all relevant information, and since it is small,
it makes it easier for humans to keep the overview and to construct
an explanation. In general, the approach is generic and can be ap-
plied to any unsupervised cluster technique to make the differences
between the identified clusters explainable to humans.
In summary, our method directs the view to essential genes in a
dataset that make the differences in the measured scenario for
clearly separable phenotypes. The selection method (PFA) is novel
and thus brings a new perspective into the gene selection portfolio,
extending existing methods. The clear view of the relevant in-
formation makes the contrast of the phenotypes significantly visible
A. Caliskan, D. Caliskan, L. Rasbach et al. Computational and Structural Biotechnology Journal 21 (2023) 3293–3314
3295
and thus better understandable for humans. Since we provide the
code of our entire pipeline via a GitHub repository, starting from the
preprocessing and ranging to the ML-validation part, the application
is immediate, and the pipeline can be easily extended in the future.
The rest of this work is organized as follows: We explain our
pipeline in detail in the Methods section and provide the results
from the analysis of publicly available single-cell data derived from
human white adipose tissue [18] and human thoracic aorta [32] in
the Results section. In the Discussion, we explain variants of our
pipeline and future research in which the pipeline can be used.
Possible applications include analyzing the differences between cells
that are resistant to therapy and those that react to a specific
treatment or potential therapy approach. Further applications in-
vestigate the effect of therapy on resistant cells and the differences
between resistant cells before and after therapy. The Conclusion
summarizes the advances and future perspectives of our work.
2. Methods
In this section, we describe the components of our software pi-
peline that is available via a GitHub repository under https://github.
com/AC-PHD/Seurat_PFA_pipeline. Fig. 1 shows an overview of how
the different scripts are used sequentially and what their purpose is
for the analysis pipeline.
Using two different publicly available Single Cell datasets (see
Supplementary Table 1 “datasets”), we tested our software pipeline.
We chose the “single-cell atlas of human and mouse white adipose
tissue” (WAT), published by Emont et al. in 2022 [18] and available
via the Single Cell Portal (https://singlecell.broadinstitute.org/
single_cell), as a Seurat object (in RDS-file format) [13–16] to train
and test the software.
To verify the promising results obtained from this dataset, we
used a second data-set, the Single Cell data generated during the
study “Deep learning enables genetic analysis of the human thoracic
aorta” by Pirruccello (2022) [32], which is available in the h5ad-
format via the Single Cell Portal.
The first requirement is a Seurat [13–16] framework for pre-
processing the single-cell dataset. The preprocessing works as
follows.
2.1. Data preprocessing in R
The first Single Cell Portal dataset was available as an RDS-file.
After loading the Seurat object containing all cells of the WAT into an
R session, we filtered it according to cell type and tissue type using
the information in the Seurat object’s metadata, as the authors of the
Seurat object identified several cell types, including endothelial and
mesothelial cells, adipose stem and progenitor cells (ASPCs), and
adipocytes [18]. Furthermore, since all of the cell types contained
Fig. 1. Workflow including data preparation, randomization, PFA gene selection and validation. The workflow figure was created using BioRender.
A. Caliskan, D. Caliskan, L. Rasbach et al. Computational and Structural Biotechnology Journal 21 (2023) 3293–3314
3296
cells isolated from adipose tissue and only some contained cells
isolated from the non-adipocyte stromal-vascular fraction (SVF) of
the adipose tissue [18], we filtered our cell types of interest also by
tissue type.
Each cell type of interest (adipocytes, ASPCs, endothelial and
mesothelial cells as well as “Fibroblast I” and “Macrophage” of the
second dataset) was subsequently filtered for quality as re-
commended in the literature, e.g., by Luecken and Theis (2019) [33].
By removing cells with a low count depth, a high number of mi-
tochondrial counts, and a small number of detected genes, we re-
moved cells with broken membranes and dying cells [33].
Subsequently, we removed doublets, i.e. two cells sequenced to-
gether as one cell, using the R package DoubletFinder following the
recommended workflow [17].
Using these results, the Seurat object was filtered. Only the re-
maining cells were used for the subsequent workflow. As the Seurat
object is filtered using tags generated during the respective analyses,
it is technically possible to use other doublet finding tools such as
the Jupyter Notebook and Python-based tools Scrublet (Single-Cell
Remover of Doublets) [34] or DoubletDetection [35], although this
might possibly disrupt the workflow. Considering that the data
preparation workflow is conducted in R, using an R-based tool such
as DoubletFinder [17] or DoubletDecon [36] would be more con-
venient for doublet detection.
Finally, the resulting “cleaned and filtered” Seurat objects were
transformed into count tables, the required input for the ML work-
flow. The counts of the Seurat object were saved as data.frame,
creating a column containing the row names. As the last preparation
step, we removed all rows containing ribosomal and mitochondrial
genes and added a label row to the data.frame. This label is required
to mathematically distinguish between the different cell types
during the PFA and machine learning process.
Since we compared two cell types as a class and thus did not
require detailed sample information, we used either 0 or 1 as the
label, removing the Seurat sample names. After preparing one cell
type as a table labeled with 0 and another as a table labeled with 1,
we merged both tables by gene name.
2.2. Data preprocessing in Python
Subsequently, the input datasets for the ML pipeline were pre-
pared using our Python script “Randomizer”. With the input data
table as input, this script selects 1000 random cells of each cell type
(e.g., ASPCs and adipocytes, also referred to as each group) from the
pool of all single cells that are given after preprocessing, creating a
“results” file (“results_1” to “results_n”, with n being the last run of
the randomizing script). The number of cells can be adjusted by
changing the variable “cell_count” in the Randomizer script (1000 is
set as the default). The remaining cells of the two groups that were
not selected from the pool for the results file are saved as “rest”
(“rest_1” to “rest_n”). This process is repeated until there are less
than 1000 cells of one of the two groups left (n times, depending on
the size of the dataset), using no single cell twice. Thus, a table
containing 4200 cells of one cell type (the first group, labeled as “0”)
and 3900 cells of the other cell type (the second group, labeled as
"1") after cleaning and filtering would result in a pool set containing
8100 single cells. However, with a Randomizer setting of 1000 cells,
it can only yield three results-files, which equals three data sets
containing 1000 cells of each of the two cell types for subsequent
analysis. This would be sufficient for three PFA analyses.
This step can be rather time-consuming, depending on the size of
the input data table and the machine used. For reference: on a PC
with 83 AMD Ryzen 9 3900X, 12-Core Processor, 64 GB RAM, 64-Bit-
Operating System, and an x64–84 based processor, using a VM Linux
environment (Ubuntu 20.04.2 LTS (OS-Type: 64-bit) using Virtual
Box 6.1) and on a Macbook Air, 2022, M2, 8 GB RAM, this calculation
took about a day for the datasets generated using the single-cell
atlas of human and mouse white adipose tissue [18]. Smaller data
sets, such as the one generated using “Fibroblast I” and “Macro-
phage” of the human thoracic aorta data [32], require less time (six
to eight hours).
2.3. Principal feature analysis (PFA)
Next, the preprocessed data is analyzed using the PFA [4] to
identify the genes required to distinguish between the cell types
(labels). This is a new concept we present here: We objectively de-
termine which selection of genes separates given clusters of cells
best from each other (or more general: which selection of features
separates given clusters best from each other). In the paper [4] the
mathematical properties of our PFA and the mathematical applica-
tion areas ranging from advantages for classical modeling to re-
gression analysis and analysis of differential equations and machine
learning are given.
More specific, we use Algorithm 2 of Breitenbach et al. (2022) [4]
where the maximum number of nodes per subgraph (called clus-
ter_size in the implementation) controls the size of the dependence
graphs that are dissected to reduce redundancy. Too big and too
small values compared to the number of total genes can cause long
calculation times. In order to discretize the continuous expression
levels of genes, we use Algorithm 4 of Breitenbach et al. (2022) [4]
where the parameter for the minimum number of data points in a
bin (called min_n_datapoints_a_bin in the implementation) controls
how many data points (measurements of a gene expression;
equivalent to the number of measured single cells) form a discrete
event meaning “the expression level of this gene in this single cell is
between the number of the measurement with the lowest and the
highest gene expression of measurements forming this bin”. Once
the expression levels of each gene are discretized into events, we can
formulate the contingency table for the chi-square test of in-
dependence where we compare two genes to see if their expression
is independent of each other. We note that the label is handled with
the same discretization algorithm to define the events like “this
measurement belongs to a single cell labeled with 0”. The higher the
min_n_datapoints_a_bin is, the coarser the discretization of the
continuous gene expression is, which could delete information as
well. However, if it is too small, then too few data points could be in
some events which could result in some expected frequencies in the
entries of the contingency table (meaning that gene A has expression
level x and gene B has expression level y) being below 5 data points;
this means that this combination of “gene A has expression level x
and gene B has expression level y” is expected in only 5 single cells,
violating the requirements for a reliable test of independence. The
concrete choices of the parameters are given with the data sets in
the Results section.
With the same discretization, we can calculate the mutual in-
formation between the expression of a gene and the label of the
single cells. The mutual information provides a measure of how
much we know about the label if we know the expression level of a
gene (and vice versa).
Let the discretized gene expression of gene
i
be modeled by the
random variable
Xi
where
Zi
is the space of bins in which the ex-
pression of the gene
i
is discretized with
z Z
k i
,
…k m{1, , }
i
,
mi
. Analogously for the label that is modeled as the random
variable
Y
with
z Z
l Y
,
…l m{1, , }
Y
,
mY
. In our example with a
label of 0 or 1,
=m2
Y
. Now, let
=P X z( )
i k
be the probability that
among all single cells gene
i
is expressed at a level assigned to the
bin
zk
(number of data points in that bin divided by the number of all
single cells) and let
=P Y z( )
l
be the probability that a single cell is
labeled with
zl
(number of single cells with label
zl
divided by all
single cells). Furthermore, let
= =P X z Y z( )
i k l
be the probability
that a measurement of gene
i
being expressed at a level assigned
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with
zk
and the corresponding single cell is label with
zl
(all gene
measurements where
=X z
i k
and the single cell is labeled
=Y zl
divided by the number of all single cells). Then, we can define the
mutual information
I
between the discretized expression level
Xi
and
Y
as follows
= = =
= =
= =
= =
I X Y P X z Y z P X z Y z
P X z P Y z
( , ) ( )log ( )
( ) ( ) .
ik
m
l
m
i k l
i k l
i k l
1 1
iY
An illustration of why the mutual information measures the re-
latedness of two random variables is the following:
= = = =
= =
=
P X z P X z Y z P X z Y z
P X z
( ) ( | ) ( )
( )
i k i k l i k l
j l
where the first equality sign holds if the expression of a gene is in-
dependent of the kind of single cell and the second one due to de-
finition. The total equation defines stochastic independence of two
random variables. The conditional probability
= =P X z |Y z( )
i k l
is
defined by the right hand-side of the equality sign “ ”. If the quo-
tient
= =
= =
P X z Y z
P X z P Y z
( )
( ) ( )
ik l
i k l
is close to 1, it means that the equality is fulfilled
and
= =
= =
log 0
P X z Y z
P X z P Y z
( )
( ) ( )
ik l
i k l
. If the equation holds for any
…k m{1, , }
i
and
…l m{1, , }
Y
, it means that the two random variables are sto-
chastically independent of each other and the knowledge of a value
of one random variable (expression level of a gene) does not allow
reliable conclusions about the value of the other random variable,
which is the label of the cell, meaning which cell type we have. In
this case the mutual information takes the minimum 0. In all other
cases, the mutual information is > 0 and weights the correlation of
the outcomes of the two random variables.
After describing our mathematical toolbox, we explain further
how we analyzed the data sets. We take three different sets of single
cells from the randomizer script, each containing two cell types and
1000 single cells of each type. We perform the PFA analysis on the
first set, called “PFA set”. The other two sets are left for validations,
explained later.
The genes returned by the PFA are highly correlated with the
differences between the cell types. These genes are ranked by the
mutual information they share with the label, as explained in
Breitenbach et al. (2022) [4]. The more mutual information an ex-
pression of a gene and a label have in common, the more informa-
tion about the label can be derived from the measurement of the
corresponding gene expression. In our pipeline, the mutual in-
formation between each gene and label is exported as a file named
“gene_mutual_information.csv”, containing the genes deemed im-
portant by the PFA. These genes can be used during the validation
step. However, it is also possible to use another gene set for vali-
dation of which one assumes to contain relevant information to
separate the phenotypes.
According to the ranking, we might neglect genes that only have
a low amount of mutual information, e.g., because their expression
level only contributes minorly to which cell type a cell belongs or
these genes are only important in a very small portion of single cells
to decide the cell fate. However, as explained in Breitenbach et al.
(2022) [4], this procedure is only an approximation since it is also
possible that many genes, which all have a small amount of mutual
information, can, when seen in combination, be important to de-
termine the cell type, which a suitable model would “learn” from the
data. To summarize, the ranking with the mutual information is an
approximation that worked well in our experiments. In case, the
pairwise mutual information does not work, our validation frame-
work, which we explain later, works for any set of genes that one
thinks captures the information about the difference of the different
phenotypes, e.g. if we have a hypothesis from domain knowledge or
intuition for the set of important genes, we can test if it provides
sufficient information to differentiate the cells into their clusters.
We note that we optimized the PFA script to use more than one
kernel by parallelizing the dissections of the dependency subgraphs.
This considerably reduced the calculation time up to 25% in our
experiments depending on the structure of the interaction of the
features and thus of the PFA’s dependence graph. The implementa-
tions of both versions are such that the output of the serial and
parallel version is identical. The parallel PFA script is available by a
GitHub repository via https://github.com/LauritzR/Parallel-
Principal-Feature-Analysis.
The serial version of the PFA can be found via https://github.com/
LauritzR/Principal-Feature-Analysis. In the supplementary material,
we provide documentation where we describe the high-level dif-
ferences between the serial and parallel PFA version.
2.4. Validation
To check how much information is lost due to neglecting genes
with little mutual information and thus balance explorative power
with a small set of genes, we use the following workflow. This
workflow can be used to validate any promising set of genes from
any source that is assumed to provide sufficient information for
distinguishing between the different phenotypes.
The main idea of the validation is the following. If the expression
levels of our selected genes contain sufficient information, it should
be possible to fit a function that can classify to which cell type/
cluster/label a single cell belongs with a high accuracy based only on
the expression level of the selected genes. This function contains the
rules or explanation, i.e., what in the expression levels determines
the cluster designation. All of our analysis is performed on the “PFA
set” generated by the Randomizer script, as described above, which
is the result_1 set from the randomizer. For the validation step, we
now include the two other sets generated by the Randomizer script
on which we only fit a function (e.g., a ML model) to show that our
gene selection is not sensitive to our “PFA set” sampled from the
pool; rather, it contains generic information about the relationship
of gene expression of the selected genes and the assignment to a cell
type/label.
We would like to stress that the model we fit is not the main
result of our analysis, but rather the set of selected genes. We simply
use the supervised ML framework because it provides a well-de-
veloped toolbox to fit a function that maps input (expression level of
genes) to output (labels). Consequently, if we know a well-working
function exists which can do the mapping based on our selected
genes on each set generated by the randomizer, we conclude that
our (small) selection of genes is meaningful with regard to de-
scribing the difference between the labels (i.e., cell types or clusters).
The reverse is of course not true, meaning if the validation fails
that our selection does not contain enough information since maybe
the model training fails (e.g., disadvantageous convergence para-
meters) or the model type does not match the real dynamics that
drive the relevant gene expression (e.g., linear vs. non-linear
models).
We remark that we would like to verify the gene selection with
respect to non-sensitivity for the data set and not necessarily a
concrete model. That is why it is justified to retrain a new model on
each set based on the same genes. One possible extension might be
to train the model only on “PFA set” and perform only inference on
the other two sets, not training again from the scratch.
This validation procedure, having a set for analysis and several
sets for validation, can also be applied for other gene selection
methods, e.g. for those mentioned in the Introduction.
More specifically, we train an ML model (MLPClassfier from
Python’s sklearn) on a training dataset and predict the cell type on a
test set for each set from the randomizer, as described above, based
on different gene selections to validate our selection’s information
content. We use an 80/20 ratio split of a dataset for training and
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testing/predicting: i.e., 80% of the dataset for each set from the
randomizer are used for training and the remaining 20% are subse-
quently analyzed as test to see whether the model can correctly
distinguish the different cell types (see discussion for justification).
We start on all available non-constant genes as the input space
for the ML model. Non-constant genes mean that there is at least one
single cell where the expression level differs from the others. The
procedure on all non-constant genes serves as a control to check the
accuracy given all available information since genes are also in-
cluded that might have been sorted out by our analysis but contain
relevant information. Consequently, we assume that in this case the
accuracy should be the best possible which we should also achieve
by our gene selection assuming that we pick genes relevant for de-
scribing the difference of the investigated labels and we only remove
non-relevant genes.
Next, we train another model of the same type (MLPClassifier) on
the genes we analyzed with the PFA and exceed a certain threshold
for mutual information. This analysis is only done on the set “PFA
set”. Now, we can see how the accuracy varies depending on the
threshold for the mutual information compared to the model trained
on all genes. If the threshold was set correctly, an analysis using a
significantly lower number of genes than all of the genes contained
in the Seurat object results in a prediction with comparable accuracy
to the prediction made using all genes. Since Seurat objects can
contain several thousand genes, an analysis requiring only a fraction
of this number will result in a shorter calculation time, and thus, in
faster results for the researcher. On the other two sets, the selected
genes are kept (no adjustment of the mutual information threshold)
and the training and testing is performed. If the selection of the
genes provides a model with sufficient accuracy and is not sensitive
to the PFA set, a function should exist as well on the other data sets
based on the same gene set with a comparable accuracy to the one of
the previous function/model fit on the PFA set.
To evaluate how much better the PFA selection is compared to
randomly selected genes, we train another model of the same type
(MLPClassifier) based on randomly selected non-constant genes
where the number of selected genes corresponds to the number of
PFA-selected genes that passed the mutual information threshold.
To take the randomness of the training process into account, we
repeat each procedure twenty times and take the mean accuracy of
the corresponding models. In the case of random gene selection, for
each sweep, new genes (number equals the PFA gene; drawing with
put back) are selected from the pool of non-constant genes. We use
only the standard settings for the MLPClassifier as they worked fine
in our cases. We will see in the Results section that with the standard
setting we could achieve sufficient accuracy proving the existence of
a function being able to map gene expression of our gene selection to
labels. However, the process would also be valid for performing a
hyperparameter tuning for each model or choosing the best model
type in each case since we only need the existence of a function to
prove the information content of our selection.
The complete workflow is summarized in Fig. 1, which was cre-
ated using BioRender (https://biorender.com/, accessed on 01
April 2023).
As explained in Breitenbach et al. (2022) [4], we cannot conclude
that the gene selection is bad if the accuracy is bad, since it is pos-
sible that the corresponding ML model is not appropriate to capture
the functional relations. However, we argue that if we have one
model that has an accuracy sufficient for the use case, we can con-
clude that our gene selection captures the relevant information.
2.5. Multiple cell types validation
As a final step, we demonstrate that our method is also capable of
discerning between more than two different cell types. Therefore,
we generated a validation dataset containing three cell types and a
validation data set containing four cell types, using 1000 cells of
each cell type. For the dataset containing three cell types, we used
ASPCs, adipocytes and mesothelium (“0”, “1″, and “2”, respectively),
for the dataset containing four cell types, we used ASPCs and adi-
pocytes as well as endothelium and mesothelium (labeled as “0″, “1”,
“2”, and “3”, respectively).
These datasets were subsequently analyzed with our method in
order to demonstrate that our algorithm can also discern three or
four different cell types.
2.6. Biological validation
Since our analysis results in genes that are – according to our
algorithm – important for recognizing the differences between two
cell types, we treat the resulting genes as differentially expressed
genes (DEGs) or marker genes. If these genes have predictive value,
they should help analyze the differences between the cell types, e.g.,
result in pathways or biological processes. Thus, we employed
clusterProfiler [37,38] (version 4.4.2), in combination with the R-
packages enrichplot [39] (version 1.16.1) and ggplot2 [40] (version
3.3.6) to visualize biological processes related to the genes resulting
from our analysis (subsequently also referred to as “PFA genes”). The
information on the GO biological processes, which is necessary for
the enrichment analysis, was obtained via the Molecular Signatures
Database (MSigDB) [41–45] (version 7.5.1). The results were visua-
lized as bar plots and CNET plots.
Additionally, we analyzed the respective cells of the Seurat object
using the standard Seurat workflow [13–16] to find marker genes:
genes which are differentially expressed between the two cell types.
To validate our PFA genes, we used three of the different testing
options available via the Seurat FindMarkers() function: DESeq2,
Wilcoxon Rank Sum test, and Student’s t-test.
3. Results
First, we provide the results of our software pipeline exemplified
with several datasets (see Supplementary Table 1 datasets). Each
data set consists of several thousand single cells of each type.
The Seurat parameters in the pipeline were set to use only cells
with more than 800 molecules per cell (nCount_RNA), more than
500 genes (nFeature_RNA), and less than 5% mitochondrial DNA
(named mt.percent or percent_mito, depending on the dataset).
Additionally, we removed all ribosomal and mitochondrial genes as
described above.
After preprocessing, each data set with all single cells (with label
either 0 or 1) is called the pool set for the corresponding cell types.
For the subsequent analyses, the pool set (e.g., the set containing
ASPCs and adipocytes) is randomly split into further subsets using
the Randomizer script, as described in the Methods section, re-
sulting in subsets containing 1000 single cells of each type, 2000
cells in total per file (“result_1″ to “result_n”).
The training set from each subset consists of 80% of all single cells
of each type, resulting in training data consisting of 800 cells of each
cell type. The test set contains the remaining 20% of single cells.
The set “Set PFA” was randomly selected from the pool set (“re-
sults_1″) and used for the PFA analysis to select the genes. The sets
“Set 1″ and “Set 2″ are additionally randomly sampled from the pool
set (“results_2″ and “results_3″, respectively), and no single cell is
used in more than one subset.
These sets are used to show that the PFA gene selection (done
only on the “PFA set”) is not sensitive to a particular set but has
general predictive power regarding the cell types. Consequently, the
gene selection from “Set PFA” is used to train and test models on “Set
1″ and “Set 2″. Furthermore, taking the gene selection from one set
and applying it to sets from the pool shows that the gene set not only
contains sufficient information, but also that the chosen sets with
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1000 single cells of each type were large enough to present the
statistical properties of the pool set.
The PFA parameters were set to
“calculate_mutual_information=True”, “cluster_size= 300″, “min_-
n_datapoints_a_bin= 100″, and to a mutual information that was
sufficient to achieve at least 98% accuracy with as few genes as
possible (mutual information > 0.5 for ASPCs and adipocytes (13 PFA
genes) as well as endothelium and mesothelium (10 PFA genes); or
0.6 for adipocytes and mesothelium, resulting in 12 PFA genes), for
analyzing different cell types from the single-cell atlas of human and
mouse white adipose tissue [18]. If not further noted, we used the
default values of the methods.
The calculation time for the PFA and validation procedure was
approximately 2–5 h, depending on the data set.
3.1. Analyzing adipocytes and ASPCs as an example
The first pool data for demonstrating our pipeline was generated
using the single-cell data on human white adipose tissue (WAT)
made available by Emont et al. (2022) [18] via the Single Cell Portal.
We used ASPCs and adipocytes derived from human WAT, labeled
with the tissue type “adipose” according to the metadata of the
Seurat object, and combined the data as described in the method
section to a table containing both cell types, labeled as cell type “0”
and cell type “1”, respectively. According to the visualization of the
study (“A single cell atlas of human and mouse white adipose tissue”) in
the Single Cell Portal, both cell types are clearly distinguishable in a
UMAP plot (Fig. 2).
PFA of a subset (“results_1″, containing 1000 cells of each cell
type) required only 13 genes (“PFA genes”) for a prediction accuracy
of almost 100% (Table 1). The accuracy of the models based on all
non-constant genes (also referred to as “all genes”) was 100% for all
three of the analyzed subsets (“results_1″, “results_2″, and “re-
sults_3″, generated by using the Randomizer script on the table
containing adipocytes and ASPCs) during training. The subsequent
tests on the remaining 20% of subsets had an accuracy between
99.5% and 100%. The (almost) zero difference between training and
testing accuracy shows that there is no overfitting of the ML model.
In addition, the high accuracy on both sets of the splitting indicates
that the gene selection contains information for deterministic rules
by which to classify the cells independent of the sampling/splitting.
To find a small set of PFA genes, we tested several thresholds of
mutual information for each analysis (Figs. 3 to 7 and Supple-
mentary Table 2 PFA results). For the analysis of ASPCs and adi-
pocytes, we evaluated the thresholds 0.1, 0.25, 0.5, and 0.6 (Fig. 3).
While the prediction accuracy using PFA genes is rather constant,
the number of wrongly classified cells using an equal number of
random genes rises with more stringent thresholds. While a
threshold of 0.1, which equals 439 randomly selected genes for the
analysis of ASPCs and adipocytes, results in a wrongly classified
mean of 26.15, a threshold of 0.25 (105 random genes) results in a
wrongly classified mean of 73.15. Using the PFA genes instead re-
sults in much lower wrongly classified means. For instance, with
only 5 PFA genes, which equals a threshold of 0.6, the wrongly
classified mean is 7.1, which is still considerably smaller than the
wrongly classified mean of analyzing 439 randomly selected genes
(26.15 wrongly classified cells) and significantly smaller than the
wrongly classified mean of analyzing 5 randomly selected genes,
which is 168.1 (see Supplementary Data “1 ASPCs & adipocytes
PFA” for the analysis of the first subset (“results_1”) and Fig. 3A and
Supplementary Table 2 PFA results).
In the plots (Fig. 3) we see that the prediction accuracy of random
genes on training and test sets (repeated 20 times, mean of all
sweeps is plotted) significantly drops at a threshold value of 0.25
(145 genes). However, when using the PFA genes, the accuracy starts
to drop between threshold values of 0.5 (13 genes) and 0.6 (5 genes).
Above the threshold of 0.5, the prediction accuracy drops below the
minimum criterium of 98%, which means that a gene whose
Fig. 2. Screenshot of the cluster visualization of adipocytes (red) and ASPCs (blue) available at the Single Cell Portal via this link, accessed on December 13th, 2022.
Table 1
Prediction results based on PFA gene selection, all genes, and random genes for ASPCs and adipocytes, using the ideal PFA threshold of 0.5, which results in 13 PFA genes.
Set PFA Set 1 Set 2
PFA genes All genes Random genes PFA genes All genes Random genes PFA genes All genes Random genes
Train 0.99246875 1.0 0.6476875 0.996125 1.0 0.68415625 0.9934375 1.0 0.65375
Test 0.997625 0.9975 0.621375 0.990125 1.0 0.650875 0.9975 0.997 0.648375
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expression level has a significant impact on cell type cluster as-
signment is missing. Consequently, our gene set should contain that
gene carrying important information for the difference. On the other
hand, such genes might be interesting for experiments since a
knockout or knockdown of their expression might prevent expres-
sing the phenotypic characteristics of cells, which could be a re-
sistance against a therapy or the transition to a different cell state.
Thus, in this example, the four genes (ranked 1st to 4th according to
PFA) found using the “threshold below the drop” might be vital for
the differences between adipocytes and ASPCs, while the other PFA
genes (the genes ranked 5th to 13th according to PFA) might be of
interest for predicting changes or differences between the cell types.
Summarizing, taking the genes directly before the accuracy on the
test set drops, is a practical way to determine a reasonable size for
the gene set.
Using only the 13 genes selected by the PFA algorithm (also re-
ferred to as “PFA genes”), with the ideal threshold of 0.5 (see Table 1,
indicated by the green line in Fig. 3B and C with the standard de-
viation being shown as green shadow (generated by repeating
training and testing on different data splits), and Supplementary
Data), resulted in an accuracy of at least 99.25% and up to 99.61%
during training and a prediction accuracy of up to 100% in testing
(with a minimum accuracy of 99.7%).
When training with all non-constant genes, the accuracy was
100% on the training and 99.75% on the test dataset, while using 13
randomly selected genes instead of the PFA genes caused the accu-
racy to drop to 64.77% on the training and 62.14% on the test dataset
(Table 1). Using this gene selection for two different sets generated
from the same pool produced comparable results (Table 1, Set 1 and
Set 2). Decreasing the threshold for mutual information and thus
including more than 13 genes increases the accuracy up to 100% as
expected when including genes that might explain corner cases and
thus might be ranked lower.
To prove the efficiency of our PFA, we repeated the same training
and testing using 13 random genes. The significantly lower accuracy
indicates that the 13 PFA genes have more predictive power than 13
randomly selected genes.
3.2. Validation: accuracy of PFA for analyzing other cell types
Additionally, we verified the algorithm using several cell type
comparisons derived from the human WAT data set by Emont et al.
(2022) [18].
The prediction accuracy of our MLP classifier of the sklearn fra-
mework on the training and test set based on different gene selec-
tions is summarized in Table 2; the detailed analyses of the
respective datasets (adipocytes and mesothelial cells, endothelial
and mesothelial cells, hAd1 and hAd2 of the dataset by Emont et al.
(2022) [18] as well as “Fibroblast I” and “Macrophage” of the dataset
by Pirruccello et al. (2022) [32]) are available in the
Supplementary Data.
For the analysis of adipocytes and mesothelium, a threshold of
0.6, equaling12 PFA genes, was ideal (Fig. 4 and
Supplementary Data).
Comparing endothelium and mesothelium, which were also de-
rived from the single cell atlas of human and mouse white adipose
tissue by Emont et al. (2022) [18], yields similar results (Fig. 5): 10
PFA genes (a threshold of 0.5), were sufficient for a prediction ac-
curacy of 100% during test.
Fig. 3. Accuracy of the prediction results of the PFA gene selection for analyzing ASPCs and adipocytes. (A) Relation of the number of selected genes (blue line) and the mean of
wrongly classified single cells (green line using the genes predicted by PFA, red line using an equal number of randomly selected genes). (B) Differences between training (blue
line) and test (green line, the green “shadow” indicates the standard deviation repeating training and testing of the models on different data splits) when using different accuracy
thresholds and the resulting PFA genes. Here a threshold of 0.5, which results in 13 PFA genes, yielded the best result for analyzing ASPCs and adipocytes. (C) Differences between
training (blue line) and test (green line, the green “shadow” indicates the standard deviation) when using the same number of random genes instead of the respective PFA genes.
The random genes result in a considerably lower accuracy (for a threshold of 0.5: 62.14% in test using 13 random genes compared to an accuracy of 98.23% in test using the 13 PFA
genes).
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3.3. Validation: using PFA to analyze the same cell type
For further validation, we compared subtypes of the same cell
type (adipocytes hAd1 and hAd2, also part of Emont et al.’s data
[18]). For the same cell type analysis, a mutual information value
of > 0.05 was used, which resulted in 249 PFA genes but yielded an
accuracy of less than 87% (86.27%) in test using the PFA
genes (Fig. 6).
This result was to be expected because in the UMAP plot these
two clusters do not separate well since they might have a similar
expression pattern (see Supplementary Fig. S3 in the Supplementary
Analyses). Notice also that the prediction accuracy on all non-con-
stant genes was not significantly better, which could be interpreted
that the gene expression data might not contain the relevant in-
formation to differentiate between the cell subtypes.
Considering the obvious overlaps of the two subclusters hAd1
and hAd2 (see Supplementary Data, Fig. S3), which becomes even
more evident when viewing the interactive online visualization at
Single Cell Portal, it is clear that the two subclusters share some
similarities, possibly resulting in an overlapping expression pattern.
This overlap could indicate that the label or the clustering (in dif-
ferent cell types/phenotypes) based on a genetic level is ambiguous.
Therefore, a low prediction accuracy was to be expected since maybe
due to a high noise or the recorded gene expression does not allow a
clear deterministic relation between the two cell types.
3.4. Validation: using PFA to analyze a second dataset
To rule out the possibility that the observed high accuracy was
specific to the data of this study [18] and to further validate our
method, we generated another pool dataset with single-cell data
published by another research group (“Deep learning enables genetic
analysis of the human thoracic aorta” [32]) who provided their results
in the h5ad-format. After converting the file into a Seurat object, we
performed the same filtering steps as described above, creating
Seurat objects for the cell types “Fibroblast I” and “Macrophage”.
Since both subsets contained cells from the ascending aorta and the
descending aorta, we split the subsets according to their ontology
label before performing the filtering and the DoubletFinder work-
flow. Due to the relatively small number of macrophages, we created
the pool data table using both ontologies, labeling all macrophages
(derived from the ascending aorta and the descending aorta) as “0”
and all cells of the “Fibroblast I” subset (derived from ascending and
descending aorta) as “1″. Fig. 4 in the Supplementary Data shows the
visualization of the two subsets (subset by cell type like our pool
dataset).
Fig. 4. Accuracy of the prediction results of the PFA gene selection for analyzing adipocytes and mesothelium. (A) Relation of the number of selected genes (blue line) and the
mean of wrongly classified single cells (green line using the genes predicted by PFA, red line using an equal number of randomly selected genes). (B) Differences between training
(blue line) and test (green line, the green “shadow” indicates the standard deviation generated by repeating training and testing on several data splits) when using different
accuracy thresholds. Here a threshold of 0.6, which results in 12 PFA genes, yielded the best result for analyzing adipocytes and mesothelium. (C) Differences between training
(blue line) and test (green line, the green “shadow” indicates the standard deviation) when using the same number of random genes instead of the respective PFA genes. The
random genes result in a considerably lower accuracy (for a threshold of 0.06: 63.20% in test using 12 random genes compared to an accuracy of 100% in test using the 12 PFA
genes).
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Using single cell data derived from another data set yields similar
results, indicating the utility of the PFA prediction for multiple types
of data sets (Fig. 7 and Supplementary Data). For the analysis of
“Fibroblast I” and “Macrophage”, a threshold of 0.5 (105 PFA genes)
resulted in a test accuracy of 100%, whereas 105 random genes re-
sulted in a test accuracy of less than 90% (89.05%). Even with a
threshold of 0.9 (4 PFA genes), both test and training accuracy were
greater than 98% (99.75% for test and 98.92% for train) when using
the PFA genes, while the test and train accuracies obtained using 4
random genes were around 55%.
3.5. PFA validation summary
Table 2 summarizes the PFA results for the different analyses.
As in Table 1, the prediction results based on the PFA gene se-
lection in Table 2 are provided under “PFA genes”. The prediction
accuracy based on all non-constant genes (meaning there is at
least one single cell where the expression level differs from the
others) is provided under “All genes”. This accuracy is provided as
a control to estimate the highest possible accuracy on a dataset.
The “Random genes” are the results based on randomly selected
genes among all non-constant genes. The number coincides with
the number of the PFA genes and delimits the PFA results from
guessing genes.
When analyzing the same cell type (the adipocytes hAd1 and
hAd2 of the human WAT data by Emont et al. (2022) [18]), a much
greater number of genes (249 genes) was required for analysis. This
resulted in a maximum accuracy of 100% in train and 86.27% in test,
indicating the similarity between both adipocyte subtypes.
The pool data generated using the aorta cells of the second study
[32], contained 33486 genes. However, our prediction method
achieved up to 99.75% accuracy using only four genes.
In Tables 1 and 2, the prediction results based on the PFA gene
selection are provided. The accuracy scores in the tables are the
mean of twenty training sweeps since, besides the random selection
of genes, the training procedure of the MLP classifier also includes
random steps, where in each sweep a standard model is trained until
convergence.
The mutual information files containing the names of the genes
used during the respective predictions are available for all experi-
ments in the supplementary information as correspondingly named
CSV files, e.g., “gene_mutual_information_ASPCs_adipocytes.csv”.
All pool datasets derived from the single-cell data on human
white adipose tissue published via the Single Cell Portal by [18]
contained 31328 genes and the pool data generated using the aorta
cells of the second study [32] contained 33486 genes. However, our
predictions only required a fraction of these genes to achieve a
prediction accuracy of at least 98.92% when analyzing two different
cell types (Table 2). The prediction analyzing ASPCs and adipocytes
required 13 genes, the prediction analyzing adipocytes and me-
sothelium required 12 genes, and the prediction analyzing en-
dothelium and mesothelium required 10 genes.
When analyzing the same cell type (the adipocytes hAd1 and
hAd2), a much greater number of genes (249 genes) was required for
analysis. This resulted in a maximum accuracy of 100% in train and
86.27% in test, indicating the similarity between both adipocyte
subtypes.
Fig. 5. Accuracy of the prediction results of the PFA gene selection for analyzing endothelium and mesothelium. (A) Relation of the number of selected genes (blue line) and the
mean of wrongly classified single cells (green line using the genes predicted by PFA, red line using an equal number of randomly selected genes). (B) Differences between training
(blue line) and test (green line, the green “shadow” indicates the standard deviation generated by repeating training and testing on different data splits) when using different
accuracy thresholds. Here a threshold of 0.5, which results in 10 PFA genes, yielded the best result for analyzing endothelium and mesothelium. (C) Differences between training
(blue line) and test (green line, the green “shadow” indicates the standard deviation) when using the same number of random genes instead of the respective PFA genes. The
random genes result in a considerably lower accuracy (for a threshold of 0.5: 58.52% in test using 10 random genes compared to an accuracy of 100% in test using the 10 PFA
genes).
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The pool data generated using the aorta cells of the second study
[32] contained 33486 genes. However, our prediction method
achieved up to 99.75% accuracy using only four genes.
3.6. Multiple cell types validation
Performing the same analysis steps using a dataset containing
three or four different cell types, resulted in a comparable accuracy.
For three cell types (1000 cells each of ASPCs, adipocytes, and me-
sothelium), a threshold of 0.38 resulted in 43 PFA genes and an ac-
curacy of 99.06% for the training data and 99.63% for test data. This
has a comparable accuracy as using all genes for the calculation
(100% accuracy for train, 98.93% for test). Using 43 random genes
instead of the selected PFA genes, resulted in an accuracy of 73.73%
for train and 55.22% for test.
Analyzing four cell types at the same time (1000 cells each of
ASPCs, adipocytes, endothelium, and mesothelium) resulted in a
threshold of 0.35 and 51 PFA genes for an accuracy of 99.64% for train
and 98.94% for test. This has a comparable accuracy as using all
genes for the calculation (100% accuracy for train, 99.23% for test).
Using 51 random genes instead of the 51 PFA genes caused the ac-
curacy to drop to 69.11% for train and 51.35% for test. The detailed
PFA results are available in the supplementary data.
3.7. Biological validation
We used several approaches to further validate our method and
the resulting genes, which are also referred to as PFA genes. To va-
lidate the PFA genes, which were obtained by analyzing a subset
containing 1000 cells of each cell type, we performed a PFA of the
complete dataset containing all cells instead of only 1000 cells of
each of the two cell types. Additionally, we analyzed the complete
dataset using the Seurat test options DESeq2, Wilcoxon Rank Sum
test, and Student’s t-test.
3.7.1. Validating the PFA genes
The original PFA dataset was created using the R-packages Seurat
[13–16] and DoubletFinder [17], following the respective preparation
steps previously described [14,17]. For the validation, we applied the
same steps, but instead of keeping only the counts and replacing the
cell names with the labels 0 and 1 for the respective groups, we
merged the Seurat objects after removing the doublets. Thus, instead
of a table containing the counts of the two analyzed groups (adi-
pocytes and ASPCs in this example) and 0 and 1 instead of the in-
dividual cell names, we received a Seurat object containing all the
information of the two cell types. This Seurat object, containing the
same cells as the complete PFA table, was subsequently analyzed.
The dimensional reduction plots generated using the Seurat Package
Fig. 6. Accuracy of the prediction results of the PFA gene selection for analyzing two cells of the same cell type (hAd1 and hAd2, different subtypes of human adipocytes). (A)
Relation of the number of selected genes (blue line) and the mean of wrongly classified single cells (green line using the genes predicted by PFA, red line using an equal number of
randomly selected genes). (B) Differences between training (blue line) and test (green line, the green “shadow” indicates the standard deviation generated by repeating training
and testing on different data splits) when using different accuracy thresholds. Here, a threshold of 0.07, which results in 249 PFA genes, yielded an accuracy of 86.27% in test. (C)
Differences between training (blue line) and test (green line, the green “shadow” indicates the standard deviation) when using the same number of random genes instead of the
respective PFA genes. The random genes result in a lower accuracy (for a threshold of 0.07: 73.28% in test using 249 random genes compared to an accuracy of 86.27% in test using
the 249 PFA genes).
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Fig. 7. Accuracy of the prediction results of the PFA gene selection for analyzing “Fibroblast I” and “Macrophage” of the dataset by Pirruccello et al. (2022) [32]. (A) Relation of the
number of selected genes (blue line) and the mean of wrongly classified single cells (green line using the genes predicted by PFA, red line using an equal number of randomly
selected genes). (B) Differences between training (blue line) and test (green line, the green “shadow” indicates the standard deviation generated by repeating training and testing
on several splits of the data) when using different accuracy thresholds. Here a threshold of 0.5, which results in 105 PFA genes, yielded the best result for analyzing fibroblast and
macrophage. However, even a threshold of 0.9, which uses only four genes, still results in an accuracy of about 99%. (C) Differences between training (blue line) and test (green
line, the green “shadow” indicates the standard deviation) when using the same number of random genes instead of the respective PFA genes. The random genes result in a
considerably lower accuracy (for a threshold of 0.5: 89.05% in test using 105 random genes compared to an accuracy of 100% in test using the 105 PFA genes, and even greater
differences for a threshold of 0.9: an accuracy of 55.81% in test using 4 random genes, compared to an accuracy of 99.75 in test using 4 PFA genes).
Table 2
Prediction accuracy summarized for each experiment.
Number of
required
genes
Set PFA Set 1 Set 2
Accuracy
using PFA
genes
Accuracy using
random genes
Accuracy
using all
genes
Accuracy
using PFA
genes
Accuracy
using
random
genes
Accuracy
using all
genes
Accuracy
using PFA
genes
Accuracy
using
random
genes
Accuracy
using all
genes
ASPCs and adipocytes (31328 genes in the input table, threshold = 0.5)
Train 13 0.99246875 0.6476875 1.0 0.996125 0.68415625 1.0 0.9934375 0.65375 1.0
Test 13 0.997625 0.621375 0.9975 0.990125 0.650875 1.0 0.9975 0.648375 0.997
adipocytes and mesothelium (31328 genes in the input table, threshold = 0.6)
Train 12 0.9945625 0.652214285714286 1.0 0.9961875 0.6823125 1.0 0.995375 0.6889375 1.0
Test 12 1.0 0.632 1.0 1.0 0.6425 1.0 0.995 0.66825 1.0
endothelium and mesothelium (31328 genes in the input table, threshold = 0.5)
Train 10 0.996875 0.60221875 1.0 0.99571875 0.62240625 1.0 0.99628125 0.61821875 1.0
Test 10 1.0 0.58525 1.0 0.9975 0.6125 1.0 1.0 0.599375 1.0
two samples of the same cell type (adipocytes hAd1 and hAd2) (31328 genes in the input table, threshold = 0.07)
Train 249 1.0 0.9995625 1.0 1.0 0.99971875 1.0 1.0 0.9994375 1.0
Test 249 0.86275 0.73275 0.893875 0.8415 0.7035 0.877375 0.845375 0.68425 0.862125
fibroblast and macrophage (33486 genes in the input table, threshold = 0.9)
Train 4 0.98921875 0.5604375 1.0 0.99065625 0.57578125 1.0 0.995375 0.59328125 1.0
Test 4 0.9975 0.558125 0.993875 0.9925 0.57175 0.997375 0.9925 0.587375 0.999875
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Fig. 8. Dimensional reduction plots generated using the Seurat package to analyze the complete data set (all cells of adipose tissue adipocytes and adipose tissue ASPCs). (A) The
Seurat object containing the complete Seurat objects for adipocytes and ASPCs (after filtering and removing the doubles as described in the methods section) contains only
adipose tissue. (B) The dimensional reduction plot visualized by individual shows no batch effects. Thus, there are no differences based on technical, non-biological factors. (C) The
dimensional reduction plot visualized by cell type shows that the Seurat object containing the complete data set (the pool data used to obtain the subsets containing 1000 cells of
each cell type) contains only adipocytes and ASPCs.
Fig. 9. Results of the FeaturePlot() visualization of the top 6 PFA genes that were found by applying a threshold of 0.5 on a randomized data set containing 1000 cells of each
group/cell type (adipocytes and ASPCs). Here, all cells of the two groups were analyzed, resulting in cell type specific gene expression. (A) According to PFA, GPAM was ranked first,
the visualization shows that its expression appears to be specific for adipocytes. (B) According to PFA, ACACB was ranked second, the visualization shows that its expression
appears to be specific for adipocytes. (C) According to PFA, CD36 was ranked third, the visualization shows that its expression appears to be specific for adipocytes. (D) According
to PFA, AQP7 was ranked fourth, the visualization shows that its expression appears to be specific for adipocytes. (E) According to PFA, LIPE-AS1 was ranked fifth, the visualization
shows that its expression appears to be specific for adipocytes. (F) According to PFA, ITGA7 was ranked sixth, the visualization shows that its expression appears to be specific for
adipocytes.
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Fig. 10. Results of the FeaturePlot() visualization of the rest of the PFA genes that were found by applying a threshold of 0.5, rank 7–13 on a randomized data set containing 1000
cells of each group/cell type (adipocytes and ASPCs). Here, all cells of the two groups were analyzed, resulting in cell type specific gene expression. (A) According to PFA, NEGR1
was ranked seventh, the visualization shows that its expression appears to be specific for ASPCs. (B) According to PFA, GYG2 was ranked eighth, the visualization shows that its
expression appears to be specific for adipocytes. (C) According to PFA, DIRC3 was ranked ninth, the visualization shows that it appears to be specific for adipocytes. (D) According
to PFA, MALAT1 was ranked tenth, the visualization shows that it appears to be higher expressed in adipocytes than in ASPCs. (E) According to PFA, HOOK2 was ranked 11th, the
visualization shows that its expression appears to be specific for adipocytes. (F) According to PFA, ADIPOQ was ranked 12th, the visualization shows that its expression appears to
be specific for adipocytes. (G) According to PFA, AL356479.1 was ranked 13th, the visualization shows that its expression appears to be specific for adipocytes. (H) Visualization of
the two different cell types for better comparison. The dimensional reduction plot visualized by cell type shows that the Seurat object contains only adipocytes and ASPCs.
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show that this validation Seurat project only contains cells from the
adipose tissue (Fig. 8A), has no batch effects and thus shows no
differences between the cell types that are based on technical, non-
biological factors (e.g., different sample handling procedures)
(Fig. 8B), and contains only adipocytes and ASPCs (Fig. 8C).
The differences between adipocytes and ASPCs can be seen in
Fig. 8C. While the adipocytes cluster closer to the top of the plot, the
ASPCs cluster closer to the bottom of the plot, indicating differences
between the expression of the two cell types. If the PFA genes allow
a valid prediction of the respective cell types, their expression
should be specific for one cell type. This is demonstrated in Fig. 8C by
the clear separation of adipocytes and ASPCs. Subsequently, we vi-
sualized the gene expression of the 13 most significant PFA genes
using the FeaturePlot() function with the respective gene name set
as feature (Fig. 9 and Fig. 10).
The gene expression of the top-ranked PFA genes appears to be
cell type specific for the nine highest ranked PFA genes (Fig. 9 and
Fig. 10), and even MALAT1 (Fig. 10D), which is expressed by adipo-
cytes and ASPCs, appears to be higher expressed in adipocytes. The
last three PFA genes (HOOK2, ADIPOQ, and AL356479.1) also appear
to be specific for adipocytes (Fig. 10).
When the complete data set is analyzed (containing all single
cells of both cell types), all cells are taken into account, a total of
24589 cells. Using the same threshold of 0.5 as for the original PFA
analysis, this results in four significant genes: CD36, ACACB, AQP7,
and ADIPOQ, which appear to be specific for adipocytes (Fig. 11) and
are among the 13 PFA genes (CD36 was ranked third, ACACB was
ranked second, AQP7 was ranked fourth, and ADIPOQ was
ranked 12th).
Since the PFA genes should be related to significant differences
between the two cell types, we validated the PFA results using GO
enrichment, treating the PFA genes as differentially expressed genes.
Performing a GO Enrichment analysis using the PFA genes resulted
in several biological processes related to metabolism and lipid sto-
rage (Fig. 12).
3.7.2. Downstream analysis of the PFA genes – GO enrichment
Since the PFA genes should be related to significant differences
between the two cell types, we validated the PFA results using GO
enrichment, treating the PFA genes as differentially expressed genes.
This analysis resulted in several biological processes related to me-
tabolism and lipid storage (Fig. 12), indicating that the two cell types
have different functions.
In their single cell atlas, Emont et al. (2022) chose the term ASPCs
(“adipose stem and progenitor cells”) to refer to cells serving as stem
cells or precursors or progenitors for adipocytes, which are also
known under a variety of other names, including Lin(-) cells, ASCs
(for adipose stem cells or adipose stromal cells), and pre-adipocytes
[18]. ASPCs are a heterogeneous cell population and can be described
as “adipocytes-to-be” [46]. A subpopulation of these cells which is
referred to as adipose stem cells (ASCs) is assumed to represent
mesenchymal stem cells that are multipotent, barely express genes
specific for adipocytes, only commit to adipogenesis after exposure
to the correct factors [46], and can be used in regenerative medi-
cine [47].
Mature adipocytes are also a heterogenous cell population
[18,46] and are involved in systemic physiology and the regulation of
the adipose depot [18]. In the post-prandial period, adipose tissue is
Fig. 11. Results of the Seurat package FeaturePlot() visualization of the four most significant PFA genes (applying a threshold of 0.5) found using the complete data for PFA
containing all cells of each group/cell type (adipocytes and ASPCs). (A) According to the complete PFA, CD36 was ranked first, the visualization shows that its expression appears to
be specific for adipocytes. (B) According to the complete PFA, ACACB was ranked second, the visualization shows that its expression appears to be specific for adipocytes. (C)
According to the complete PFA, AQP7 was ranked third, the visualization shows that its expression appears to be specific for adipocytes. (D) According to the complete PFA,
ADIPOQ was the last of the significant genes (ranked fourth, last gene before the cut off), the visualization shows that its expression appears to be specific for adipocytes.
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involved in controlling circulating fatty acids [47]. Upon insulin sti-
mulation, the uptake of fatty acids from circulating lipoproteins and
glucose and the storage of triglycerides in the adipose tissue are
increased, and lipolysis in adipocytes is inhibited [47]. In the long
term, both differentiation of preadipocytes, another ASPC sub-
population [46], and lipogenesis in adipose tissue are promoted [47].
While the visceral adipose tissue (VAT) has been associated with
obesity-related cardiometabolic diseases and inflammation, the
subcutaneous adipose tissue (SAT) might even have a protective
function [47]. Additionally, the adipose tissue metabolism might be
connected to the expression of several sirtuins, indicating their po-
tential therapeutic use for treating obesity [47]. For instance, re-
duced expression of the sirtuin SIRT6, which has been associated
with genomic stability and aging as well as lipid metabolism and
glucose homeostasis, might be involved in adipose tissue expan-
sion [47].
To see whether the next threshold (0.6), which still resulted in a
high accuracy but required only five genes, is still sufficient for
subsequent analyses, we performed GO enrichment for the five PFA
genes of the 0.6-threshold (Fig. 12). The CNET plot in Fig. 12B vi-
sualizes the top five biological processes of this analysis and the
genes associated with them.
The GO enrichment analysis was repeated using the genes re-
quired to discern between adipocytes and ASPCs when using all cells
(the complete data set, containing all single cells) and a threshold of
0.6 for PFA (Fig. 13). The CNET plot in Fig. 13B visualizes the top five
biological processes of this analysis and the genes associated
with them.
Both enrichment analyses of the PFA genes result in pathways
related to lipid storage, metabolism and energy homeostasis, which
is unsurprising and highly relevant considering the known role of
adipocytes in energy metabolism [48]. Please see Fig. 12 visualizing
the top five pathways and the genes associated with these pathways
when analyzing the genes that resulted from the PFA using 1000
cells of each cell type, and Fig. 13 visualizing the top five pathways
and the genes associated with these pathways for the analysis using
all cells of the pool data set (all cells of the adipocytes and ASCPs
instead of only 1000 each). Via the lipogenic pathway, e.g., adipo-
cytes in the WAT store excess energy in the form of triglycerides and
release glycerol and fatty acids via the lipolytic pathway [49,50].
Additionally, adipose tissue-derived factors can modulate the sys-
temic metabolism [49,50] and adipose tissue is known to regulate
energy homeostasis [51]. This indicates that both the PFA genes and
the genes resulting from analyzing the complete dataset are asso-
ciated with adipocyte specific processes, which in turn indicates the
importance of the PFA genes for discerning adipocytes and ASPCs.
The results of the enrichment analyses of the other datasets as well
as the PFA genes are available in the supplementary data (Pathways
and mutual information, respectively).
3.7.3. Comparing the PFA results to standard methods
As a last validation step, we compared our PFA genes to marker
genes found using the Seurat [13–16] function FindMarkers().
Therefore, we analyzed the Seurat data set described above using the
standard parameters of the FindMarkers() function and choosing
three analysis options: “wilcox”, the default option, which identifies
differentially expressed genes using a Wilcoxon Rank Sum test;
“DESeq2″, which implements the functionalities of the DESeq2
package [20]; and “t”, which uses the Student’s t-test.
The respective results, which are available in the Supplementary
Data (Supplementary Table 3 and the respective *.txt-files), were
filtered according to their adjusted p-values, keeping only genes
with p_val_adj < 0.05. This resulted in 2399 genes according to the
Wilcoxon Rank Sum test (Fig. 14, green), 7827 according to the
DESeq2 implementation (Fig. 14, yellow), and 2702 genes according
to the Student’s t-test (Fig. 14, red). Except for MALAT1, which was
not among the DEGs of the DESeq2 analysis, all of the 13 PFA genes
were found by all three test methods.
This demonstrates that all of the PFA genes are relevant marker
genes for distinguishing between the two cell types. That seven of
these genes are not among the most relevant genes of the standard
Seurat analysis methods indicates that our algorithm is able to re-
cognize possibly relevant genes that might have been overlooked
using the other methods. Table 3 summarizes the respective ranks of
the 13 PFA genes using the different methods.
The ranking depends on the analysis method. The different
ranking of several of the genes using different analysis methods
(Table 3) indicates that some genes resulting from PFA analysis
Fig. 12. GO Enrichment analysis of the five PFA genes obtained by performing a standard PFA (1000 cells of each group) with a threshold of 0.6. (A) Pathways related to the five PFA
genes that were selected by PFA to discern between adipocytes and ASPCs. (B) CNET Plot visualizing the linkages between the top five GO biological processes and the genes
related to them.
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might not have been among the genes of interest resulting from one
of the other methods. Other genes, such as ACACB, were among the
top genes for all analysis methods. This shows that the PFA removes
redundancy without removing relevant information.
In summary, our method is able to find a set of genes that appear
characteristic for the differences between the cell types. With a
threshold of 0.5 of mutual information between a gene expression
and the label, about half of these genes are also among the top DEGs
found by other methods. However, the other genes are also cell type
specific even though they are not among the top-rated genes when
using the other methods. This opens two possible uses: (i) As a
threshold of 0.6 still yields sufficient accuracy, not all of the genes
are required to correctly identify the cell type, which means that
reducing the number of genes might improve the time required for
Fig. 13. GO Enrichment analysis of the three PFA genes obtained via the complete PFA with a threshold of 0.6. (A) Pathways related to the three PFA genes that were selected by
PFA to discern between adipocytes and ASPCs. (B) CNET Plot visualizing the linkages between the top five GO biological processes and the genes related to them.
Fig. 14. Comparison of the PFA genes and the genes found using the FindMarkers() function of the Seurat package when analyzing adipocytes and ASPCs Comparing the top 13 PFA
genes resulting from a threshold of 0.5 (blue) and all significant up- and downregulated genes according to the Seurat FindMarkers() test option “DESeq2″ (yellow, see
Supplementary Data), all significant up- and downregulated genes according to the Seurat FindMarkers() test option “wilcox” (green, see Supplementary Data), and all significant
up- and downregulated genes according to the Seurat FindMarkers() test option “t” (red, see Supplementary Data), respectively.
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sorting the cell types. (ii) The genes that are not strictly necessary for
recognizing the cell type efficiently are still cell type specific but not
among the top genes other methods find. Therefore, these genes
might reveal interesting insights that might not have been obvious
using other methods. One example is ITGA7.
4. Discussion
In this work, we use principal feature analysis (PFA) to identify
those genes which carry the highest information suitable to separate
the groups we want to distinguish. The PFA is a powerful new ap-
proach brought in this paper to the analysis of gene selection or
biological features in general. It does not matter how many groups
we want to distinguish nor by which method the different groups
were established first. This can be done by any gene method of
choice or any combination of several methods. However, as long as
our mathematical approach has not yet been applied to these data,
we can use it advantageously to improve the analysis in identifying
the genes with the highest information content regarding separation
of the two groups.
To illustrate the power of our approach we implemented, pro-
vided and explained PFA in a software pipeline starting with gene
selection using Seurat and using PFA to judge and improve the
quality of the separation of the gene clusters established by our gene
selection method (starting with Seurat). PFA hence delivers opti-
mally separating small gene sets and hence PFA and our im-
plemented pipeline is advantageously used to analyze single-cell
data. Our pipeline identifies genes carrying information to de-
termine to which cluster/cell type a single cell belongs based on the
expression data of the genes. It allows to objectively determine cell-
type specific signatures for any type of omics data by these criteria.
We trained and tested the application with a subset of single-cell
data downloaded from the Single Cell Portal using a study that
identified different cell types in human white adipose tissue [18]. To
verify the application, we used different subsets of the study [18] as
well as single-cell data generated during another study of human
thoracic aorta [32].
Validation is necessary for our method. However, as we do not
have a clustering method but rather a method to evaluate the genes
best separating clusters established by any method of choice. K-fold
validation cannot be directly applied but the procedure has to be
modified as follows: K-fold cross validation would mean to do the
PFA selection process on each of the K sets and do the mutual in-
formation approximation on each of the K sets. Subsequently, we
could compare intersections of the results and come to an overall
gene set.
Though this is a valid selection procedure, however, each run
might not deliver a unique set of genes that each fulfils the re-
quirement of carrying the necessary information: As an example,
take two disjunct pathways with each 5 genes that are linearly
downstream connected in each pathway. The expression of the 5
genes in one pathway is correlated assuming that only the top gene
gets regulated by the environment. Then any combination of two
genes where the first one is from the first pathway and the second
one is from the second pathway is a valid result of the PFA selection
process. Depending on small variations of the (pool sampled K-fold)
data set, different combination can come out. However, each com-
bination fulfils our requirement of carrying the full information
needed to describe the expression state of the total gene network.
With a K-fold selection process it can be that we get each time a
different combination.
The validation of our pipeline is as follows: We take a part of our
data (samples), decide for a selection of genes, and if they pass the
validation, meaning carrying the information to separate cells also
on other not seen data, then we can proceed to go into a reasoning
process where the causality is in the focus and an explanation why
these genes are important for the difference from a biological per-
spective.
Our pipeline can of course be used in a K-fold cross validation by
dividing the total data set into K data sets and do our pipeline within
each auf the K data sets. However, in terms of correlation and the
example mentioned above, there is no evidence why a broad inter-
section of genes from each K-fold set indicates causality or why the
intersection has potential for a good explanation.
From our pipeline, we used a small but meaningful set of genes
on which a model could be built to classify the single cells. We used
cell types that showed a clear clustering in a UMAP plot and thus
demonstrate for the first time that our pipeline can be used to ex-
plain UMAP plots in an explainable AI scenario applied in life science.
Apart from explaining UMAP or t-SNE plots or the differences
between cell types that are defined, e.g., via markers, there are fur-
ther applications such as the following.
If we have, e.g., tumor cells in culture that are known to be re-
sistant to a specific treatment, we can split the culture and measure
the gene expression of single cells before (one split) and during/after
(the other split) the treatment. The genes resulting from our pipeline
might be the genes that are triggered by the treatment and might be
responsible for the resistance. Blocking these genes with another
drug during therapy and thus avoiding the transition to the different
gene expression program might result in a successful compound
therapy.
Another approach could be to analyze resistant and non-resistant
tumor cells with our pipeline. The differences in the expression
profiles could also hint at which genes/genetic expression program
is important for the resistance. The resulting potential therapeutic
targets might be adjusted with a drug before or during treatment.
Similarly, healthy and tumor cells can be analyzed to see differ-
ences in the expression profile.
Table 3
The 13 PFA genes comparing adipocytes and ASPCs (threshold = 0.5) compared to their rank according to Seurat analysis using DESeq2, Wilcoxon Rank Sum test, and Student’s t-
test, respectively.
Gene PFA rank DESeq2 Wilcoxon Student's t test
GPAM 1 8 upregulated 2 upregulated 2 upregulated
ACACB 2 6 upregulated 4 upregulated 4 upregulated
CD36 3 5 upregulated 6 upregulated 6 upregulated
AQP7 4 38 upregulated 16 upregulated 16 upregulated
LIPE-AS1 5 14 upregulated 7 upregulated 7 upregulated
ITGA7 6 69 upregulated 24 upregulated 24 upregulated
NEGR1 7 5 downregulated 1 downregulated 1 downregulated
GYG2 8 40 upregulated 15 upregulated 15 upregulated
DIRC3 9 13 upregulated 9 upregulated 9 upregulated
MALAT1 10 - 189 upregulated 189 upregulated
HOOK2 11 68 upregulated 29 upregulated 29 upregulated
ADIPOQ 12 43 upregulated 25 upregulated 25 upregulated
AL356479.1 13 45 upregulated 18 upregulated 18 upregulated
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The unique contribution of our pipeline is that it does not focus
on approximating the whole dataset by capturing some amount of
variance, like the principal component analysis, but directs the view
to the genes in the dataset relevant for describing the differences.
Furthermore, since we reduce redundancy, we provide a minimal
span of genes that is important for this difference. Since the set
might be much smaller than a gene set provided by a differential
expression study, it fosters experimental investigation by narrowing
down possible drug targets, for instance.
In case the important pathways in total are of interest, there are
two ways to approach this. If one knows the associated pathways of
the genes selected by our, e.g., by experience/domain knowledge or a
database, then pathways are identified. If there is no hint for path-
ways, we can use the adjacency matrix of the independence graph
assembled during the PFA to check with which genes a PFA-selected
gene is associated. Since genes within the same pathway take their
expression value not independent of each other, the adjacency ma-
trix connects genes that take their expression values not in-
dependently of each other and thus provides valuable information
for identifying pathways starting from the genes returned by our
pipeline checking to which genes a PFA-gene is connected in the
dependence graph. Please see Breitenbach et al. (2022) [4], parti-
cularly Section 2, for details about the chi-square test of in-
dependence and the resulting dependence graph with
corresponding adjacency matrix. By this procedure for identifying
pathways, we can reinduce redundancy purposefully.
The PFA is not limited to discrete labels representing the output
function as the cell types. Apart from a binary label, multi classes can
also be investigated where our pipeline provides a minimal set of
genes to cluster a single cell into one of the multi classes. The PFA
can also be used for vector-valued output functions with continuous
ranges to describe, e.g., RNA counts as a function of morphological
information, as presented in Haghighi et al. (2022) [52].
A further application of a multi-dimensional output function
with continuous values is the integrative model (Fig. 6F in Neftel
et al. (2019) [53]) where the 2-dimensional coordinates of the cells
in the plot can be used as the values for the output function of the
PFA method to obtain genes from where the transitions to the dif-
ferent cell states might be modeled.
The label row in the input data file only needs to be filled ac-
cordingly or the several first rows in the file in case of an output
vector, adjusting the number_output_functions in the PFA scripts to
the dimension of the label vector. No code needs to be adjusted.
Thus, the PFA may support the cut-off of relevant data in such big
data scenarios, which may improve model accuracy, explainability of
the models, and scalability of the data.
The rationale is that our approach is generic, not limited to purely
genetic data, and thus a strong method to identify the key features of
complex datasets and reduce the complexity of large-scale datasets
while identifying the key gene clusters/features and signatures. The
recent publication by Haghighi et al. (2022) [52] provides a case in
point.
The limitations of our pipeline are essentially the same as those
of the method used for gene selection. We applied the novel
mathematical method of PFA to biological data, focusing on single
cell sequencing data focusing on identification of gene sets re-
presenting each cluster.
Resulting mathematical limitations are discussed in detail in
Breitenbach et al. (2022) [4]. Briefly, the main point is that if the
dataset is missing independent information, e.g., the expression of
some genes or DNA segments, respectively, is not measured and
included in the dataset, then important information can be lost by
removing genes from the dataset that carry important information
for classifying the single cells. Mathematically spoken, if not all re-
levant arguments of a function are in the data set, the function that
is removed could have had some information to restore the
information of the missing argument in some cases. However, with
our pipeline setting, this effect has never been relevant, if it hap-
pened at all, since the accuracy of our validation step was very high,
indicating that sufficient relevant genes have been selected. Since in
the current implementation a chi-square test for independence is
used, we need to ensure that the mathematical assumptions for the
chi-square test are fulfilled, like that in each entry of the contingency
table at least 5 data points are expected. As an alternative to the chi-
square test, we can use any other statistical test which fits more to a
small number of single cells and which does not rely on the as-
sumptions of the chi-square test. However, if the number of single
cells (or in general measurements) is low, we never know how
sensitive results are when we would do the analysis on another
small data set. In the case of a small data set, we rather recommend
working on the measurement process to make a bigger data set
possible. Such an improvement is always beneficial independent of
the methods used.
Even if there is a lack of labels between which a difference in the
expression profile is to be identified, the PFA could be used. For this
purpose, the first stage of the PFA could be utilized to reduce the
dimensionality of the space that represents the single-cell mea-
surements where the expression of each gene is a dimension. The
first stage of the PFA cancels out genes that are a function of others,
leaving the relevant genes that drive the dynamic in the single-cell
dataset. As described in Breitenbach et al. (2022) [24], performing
this reduction as a preprocessing step might mitigate the curse of
dimensionality [31], facilitating the clustering with, e.g., t-SNE or
UMAP. Thus, in future research, it will be valuable to investigate how
this first part of our pipeline might contribute to clustering single
cells and thus identifying or defining cell types in sophisticated
scenarios where genes with redundant information cause a curse of
dimensionality.
More specific, the procedure could look as follows in the case of
unknown cell types. Before the UMAP or t-SNE clustering, we could
use the part of the PFA that reduces redundant genes and return only
these genes that are relevant as arguments for the measured dy-
namics within the single-cell dataset. Next, for the determined
clusters in the plane returned by the UMAP/t-SNE algorithms, a
density-based clustering could be performed like DBSCAN to auto-
matically label according to the UMAP/t-SNE clustering.
Alternatively, marking the clusters manually by defining coordinates
(guided by the UMAP/t-SNE clusters) within which data points
should get the same label is also possible. In a final step, identifying
differences in the expression levels between clusters might provide
markers to define and separate these cell types from each other on a
phenotypic level in the considered scenario. Additionally, we can use
the identified differences on a genetic as well as on an expression
level to analyze and explain if cell type definitions are reasonable,
including context information in which the single-cells are cultured
and measured.
Next, we discuss our findings from our investigated data sets,
using the top thirteen PFA genes found by analyzing adipocytes and
ASPCs. First ranked (according to PFA) was glycerol-3-phosphate
acyltransferase (GPAM), which is mainly expressed in adipose
tissue by adipocytes, according to its entry in the Human Protein
Atlas [54,55]. Acetyl-CoA carboxylase beta (ACACB), which was
ranked second by PFA, is also associated with adipose tissue and is,
according to its entry in the Human Protein Atlas, most abundant in
adipocytes [54,55]. CD36, which is also associated with adipose
tissue and adipocytes [54,55], is known to play a role in the ab-
sorption of long-chain fatty acids [56]. The fourth PFA gene, aqua-
porin 7 (AQP7), is also involved in energy metabolism and
expressed in adipose tissue [57]. The LIPE antisense RNA 1 (LIPE-
AS1), which has recently been reported as spanning the genes
Ceacam1 to Lipe and potentially playing a role in adipogenesis [58],
is primarily expressed in adipose tissue but also in adrenal and
A. Caliskan, D. Caliskan, L. Rasbach et al. Computational and Structural Biotechnology Journal 21 (2023) 3293–3314
3312
testis tissue [54,55]. The integrin subunit alpha 7 (ITGA7), which is
primarily expressed in adipose but also in heart tissue [54], is, ac-
cording to the Human Protein Atlas, expressed in several tissues
and cell type enriched in adipocytes in subcutaneous, visceral and
breast tissue [55]. The seventh PFA gene, neuronal growth regulator
1 (NEGR1), is the only PFA gene that was associated with ASPCs in
our analysis, and while it is also expressed in adipose tissue, it is
primarily expressed in the brain [54] and has only recently been
associated with a possible role in human obesity [59] and inter-
action with CD36 [60]. The glucose-metabolism-related glycogenin
2 (GYG2) [61], is highly expressed in adipose tissue [54], and ac-
cording to the Human Protein Atlas, cell type enriched in adipo-
cytes in several tissues [55]. Like GYG2, the ninth PFA gene,
disrupted in renal carcinoma 3 (DIRC3), has been associated with
renal cancer [61,62]. It is expressed in a variety of tissues, including
adipose tissue [54]. The tenth PFA gene metastasis associated lung
adenocarcinoma transcript 1 (MALAT1) is primarily expressed in
bone marrow although it is also expressed in a variety of other
tissues, including adipose tissue [54]. According to our analysis, it is
slightly higher expressed in adipocytes compared to ASPCs (Fig. 9I).
Although it is included in the PFA genes resulting from using a
threshold of 0.5, it might not carry substantial information for the
classification, since a threshold of 0.6, which results in only five PFA
genes, is sufficient for a correctness of 99% (see Supplementary
Data) where MALAT1 is not contained. However, the result might
also indicate a need for further research since the ML algorithm
also predicted LIPE-AS1, whose role in adipogenesis only recently
emerged [58] and NEGR1, whose participation in regulating the
cellular fat content was only reported last year [60]. HOOK2, which
was ranked 11th, has been found to be differentially methylated in
individuals with obesity and type 2 diabetes and might play a role
in type 2 diabetes [63]. ADIPOQ, which was identified as an adipose
gene in 1996 [64], might be a candidate gene for renal disease and
diabetes [65]. AL356479.1 has – as MALAT1 – been associated with
breast cancer [66] and appears to have significant effect on breast
cancer survival [67]. This could be due to the majority of the tissue
donors in Emont et al.’s study being female [18] and might indicate
existing or future health problems for some of the study partici-
pants, or might indicate a so far unknown function of AL356479.1 in
adipocytes. All of these genes coincide with the most relevant
genes of the PFA analysis, indicating that PFA is a reliable method
for finding potentially relevant DEGs or cell type markers and can
be used to find DEGs that might have been overlooked by using the
standard analysis methods.
5. Conclusion
In this work, we have applied a principal feature analysis (PFA) to
analyze single cell data sets to identify genes separating cell types/
phenotypes/classes (cell type-specific signatures). A software pipe-
line was developed consisting of a preprocessing Seurat framework
and an analysis part consisting of the PFA and mutual information.
The gene selection was tested with a machine learning framework to
balance accuracy and the number of relevant genes in a meaningful
small set of selected genes. A further contribution of our pipeline is
in the area of explainable artificial intelligence (AI) since the pipeline
provides foundations for understandable reasoning for the pheno-
typic difference from the identified genes. Future research will apply
the pipeline to discover even molecular disease pathways, for in-
stance those responsible for tumor cells becoming resistant to
therapies. In the example, this would be new pharmacological in-
hibitors that prevent cells from switching to a tumor resistance
program and reducing side-effects in general by better targeting the
relevant genes and their gene products.
Author’s contribution
AC performed the data analysis and the Seurat analyses vali-
dating the PFA results. AC, DC, TB implemented the code for the
analysis pipeline. DC created the Randomizer script. LR, TB worked
out the parallel version of the PFA and LR implemented and opti-
mized the corresponding version. WY was involved in the data
analysis. TB drafted the manuscript and AC, TB, TD were involved in
writing and revising the manuscript. TB drafted and worked out the
concept of the pipeline. AC prepared the data sets for the pipeline
and created the preparation workflow. TB and LR implemented the
subsequent steps. TB and TD co-supervised the study.
Declaration of Competing Interest
The authors declare that they have no competing financial in-
terests or personal relationships that could have appeared to influ-
ence the work reported in this paper.
Acknowledgement
We thank Sara Giddins for proofreading of the manuscript.
Furtermore, we thank DFG for funding (project number 492620490/
SFB1583-INF) and the Julius-Maximilians-University of Würzburg
for support by its open access publication program.
Appendix A. Supporting information
Supplementary data associated with this article can be found in
the online version at doi:10.1016/j.csbj.2023.06.002.
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