Tanaka

# Dynamic change of gene-to-gene regulatory networks in response to SARS-CoV-2
infection

Yoshihisa Tanaka  Graduate School of Pharmaceutical Sciences, Kyoto University
RIKEN Cluster for Science, Technology and Innovation Hub, Medical Sciences
Innovation Hub Program  Kako Higashihara  Graduate School of Medicine, Kyoto
University  Mai Adachi Nakazawa  Graduate School of Medicine, Kyoto University
Fumiyoshi Yamashita  Graduate School of Pharmaceutical Sciences, Kyoto
University  Yoshinori Tamada
Yasushi Okuno  RIKEN Cluster for Science, Technology and Innovation Hub,
Medical Sciences Innovation Hub Program

###### Abstract

## Abstract

The current pandemic of SARS-CoV-2 has caused extensive damage to society. The
characterization of SARS-CoV-2 prpfiles has been addressed by researchers
globally with the aim of resolving this disruptive crisis. This investigation
process is indispensable for an understanding of how SARS-CoV-2 behaves in
human host cells. However, little is known about the systematic molecular
mechanisms involved in the effect of SARS-CoV-2 infection on human host cells.
Here, we have presented gene-to-gene regulatory networks in response to SARS-
CoV-2 using a Bayesian network model. We examined the dynamic changes of the
SARS-CoV-2-purturbated networks established by our proposed framework for gene
network analysis, revealing that interferon signaling gradually switches to
the subsequent inflammatory-cytokine signaling cascades. Furthermore, we have
succeeded in capturing a COVID-19 patient-specific network in which
transduction of these signalings is coincidently induced. This enabled us to
explore local regulatory systems influenced by SARS-CoV-2 in host cells more
precisely at an individual level. Our panel of network analyses has provided
new insight into SARS-CoV-2 research from the perspective of cellular systems.

###### keywords:

SARS-CoV-2 | gene network analysis | COVID-19 individual network

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## Introduction

Figure 1: Illustration of overview. The hairball (blue) is the basal network
consisting of 127,126 edges and 15,258 nodes established by use of the
respiratory viruses RNA-seq including SARS-CoV-2. The highlighted-network
(magenta) in the basal network represents the COVID-19-perturbated network
extracted by using the biopsy RNA-seq.

The newly emerging coronavirus, severe acute respiratory syndrome coronavirus
2 (SARS-CoV-2), has spread rapidly over the world (1, 2), with more than
22,000,000 cases of coronavirus disease 2019 (COVID-19) and 790,000 deaths as
of August 21, 2020 (3). This pandemic outbreak has drastically changed our
society, and has compelled us to stay alert to the continuous risk of SARS-
CoV-2 infection (4). To overcome this dire situation, the development of novel
drugs or vaccines is still an urgent global challenge. During the therapeutic
development process, the elucidation of cellular mechanisms is essential for
the discovery of potential targets; the fundamental question to be solved is
how SARS-CoV-2 influences host cells and causes COVID-19 at the molecular
level. However, these cellular mechanisms for COVID-19 are poorly understood.

High-throughput technologies have contributed to the acquisition of a large
amount of “omics” data, which has provided comprehensive information on the
cellular systems. These technologies have also featured during the current
research into SARS-CoV-2. Several reports have provided various clues to
understanding the global cellular signatures in response to SARS-CoV-2
infection at both the proteome and transcriptome level (5, 6, 7). Recently,
network-based approaches have stimulated great interest in the use of these
emerging omics data for drug discovery and systems biological analysis in the
current SARS-CoV-2 field (8, 9, 10, 11, 12, 13). Their major approaches
combine publicly available sources, including knowledge of the already
established pathways and drugs with these omics data to reconstruct molecular
networks. However, these networks do not sufficiently represent a real-world
cellular system owing to two main reasons: 1) public data consist of
heterogeneous knowledge that has been accumulated throughout the longstanding
biological researches; and 2) the previous works use mixed networks that
combine data from various samples, but cannot reflect an individual
cell-/patient- specific network.

To address these problems, we recently developed a method to extract a core
sample-specific network from a massive gene network generated from a Bayesian
network (14). Gene regulatory network estimation has been developed as a
prospective method to model the cellular system using omics data (15, 16, 17,
18, 19). Although Bayesian network-based approach can infer the cause-and-
effect relationships between genes with transcriptome data, the key issue has
been to extract biologically significant information from the huge and
complicated network, which is often sarcastically referred to as a hairball
(20). Our unique framework consists of three steps: 1) estimation of a global
gene network; 2) extraction of context-specific core networks based on
differences in molecular systems from the global network; and 3)
identification of a sample or patient specific network (Fig. 1). The prominent
advantage is that it enables us to identify putative context-specific or
sample-specific potential sets of edges in the form of a network, i.e., gene-
to-gene relationships with directions, as well as nodes.

In this study, by using our developed framework for gene network analysis, we
have presented the core host cellular systems involved in SARS-CoV-2 over
several in vitro experiments; different viral loads, cell lines, and
respiratory viruses. No studies have been performed on the computational data-
driven gene regulatory network approach regarding SARS-CoV-2. We characterized
interferon signaling and subsequent inflammatory signaling cascades as
significantly changed networks in human host cells, which represent the innate
antiviruses-immune system in response to SARS-CoV-2 infection. In addition,
given that the recent studies have reported that patients with COVID-19
exhibit various clinical outcomes depending on each patient, and that a
certain proportion of patients will experience a severe disease (21, 22, 23),
it is much more important to reveal the cellular mechanisms causing these
clinical symptoms at an individual level. To this end, we have further
identified the gene networks specifically for patients with COVID-19. We
believe that our landscape of gene networks is beneficial to achieve an
understanding of how cellular systems respond to SARS-CoV-2 and to further
drug development.

## Results

### Estimation of basal gene network in the involvement of respiratory viruses
infection using a Bayesian network

We first characterized a global gene network (hereafter referred to as the
basal network) using a Bayesian network (see Methods) with a transcriptome
dataset involved in the engagement of respiratory virus infection, including
SARS-CoV-2, in several human cell lines (7). To determine a basal network
structure, we performed a network estimation using the neighbor node sampling
and repeat algorithm (24), and screened the best algorithm parameters for the
target dataset, as described in our previous study (14). Briefly, the network
estimation was run three times independently, and the subsequent concordance
test was performed to ensure the robustness and stability of the estimated
basal network. We confirmed that the iteration number $T=500,000$ satisfies
less than 5$\%$ error (Error = 4.0$\%$ for $T=500,000$; error = 5.3$\%$ for
$T=300,000$). The final basal network comprised 127,126 edges and 15,258
nodes, with a threshold of 0.05 and an average degree of 16.7. We used this
final basal network for the subsequent analyses.

### Dynamics of host cellular network profiles in different viral loads of
SARS-CoV-2

Figure 2: Dynamics of the SARS-CoV-2-perturbated network for different viral
loads in host cells. (A) The histograms of $\Delta$ECv for different SARS-
CoV-2 viral loads; a low MOI of 0.2 (blue) and a high MOI of 2 (magenta). The
X-axis corresponds to the threshold for each $\Delta$ECv. The Y-axis shows the
number of edges on a log scale. (B) The Venn diagram represents the numbers of
differentially regulated edges (DREs) for two SARS-CoV-2 viral loads (blue:
low MOI, magenta: high MOI) with a threshold of 1.0 for $\Delta$ECv in Fig.
2A. (C) The top 10 terms of canonical pathway analysis for the genes
comprising a union set of $\Delta$ECv-extracted DREs in the Venn diagram
analysis (Fig. 2B). (D) The whole picture for the various sizes of subnetwork
fragments is shown. Image of how the $\Delta$ECv-extracted DREs mutually
connected and generated the subnetworks.

To examine the transition of host cellular system dynamics during the increase
of SARS-CoV-2 viral loads, we aimed to characterize the networks perturbated
by SARS-CoV-2 with two viral loads; a low multiplicity of infection (MOI) of
0.2 and a high MOI of 2 in A549 cells. We expected that cells exposed to
different viral loads each present a unique cellular system, and that our
approach could capture the fluctuation of system dynamics in whole cellular
systems. To obtain differential core gene networks for each viral load, we
followed the multiple steps using an edge quantification technique, called
edge contribution value (ECv), established in our previous study (14). We
first calculated $\Delta$ECvs following the Eq. (2) where $S$ = SARS-CoV-2
infected and $T$ = mock samples for each MOI condition (see Methods). The
distributions of $\Delta$ECv showed that the innate cellular system was more
extensively perturbated in the cells exposed to the high MOI than the low MOI
(Fig. 2A). We next set a threshold of 1 for $\Delta$ECv and obtained
differentially regulated edges (DREs) from the basal network. The Venn
analysis for the $\Delta$ECv-extracted DREs showed that the number of DREs in
the high MOI was larger than in the low MOI (Fig. 2B). Interestingly, the
number of shared DREs between high- and low- MOIs was 42, which was only 6$\%$
of the total number of DREs in both conditions, indicating that the underlying
regulatory system between them was not similar. To confirm the biological
involvement of the DREs, we performed canonical pathway analysis for the genes
contained in the $\Delta$ECv-obtained DREs, showing that these genes were
associated with some cellular antiviral systems (Fig. 2C). These results
supported that the components of the DREs were biologically relevant to viral
infection.

Figure 3: The SARS-CoV-2-perturbated host cellular network in response to
different viral loads in A549 cells (the SARS-CoV-2-perturbated network). The
network comprises 130 nodes and 305 edges (including 155 basal edges). The
colored solid edges represent the SARS-CoV-2-perturbated DREs; high MOI of 2
(magenta), low MOI of 0.2 (blue), high MOI $\cap$ low MOI (purple). Dot edges
represent the basal edges (grey). The nodes (green) represents the known drug
target genes (Supplementary Table S1). The node size stands for the extent of
outdegree.

To gain a greater insight into the profiles of the DREs from the perspective
of network topology, we next generated networks using a set of all the DREs in
the Venn diagram (Fig. 2B). These DREs connected mutually and, in turn,
generated various sizes of subnetwork fragments (Fig. 2D). We reasoned that if
these fragments represented some biological significance, these features
should be reflected as modularity, as biologically close functions in cellular
systems link together and shape modules (25). Hence, small-sized fragments
were likely to be less informative, and we focused on the biggest connected
component among the various fragments. The biggest connected component was
extracted and the basal edges were additionally mapped on this network, which
established the SARS-CoV-2-perturbated network with 130 nodes and 305 edges
(Fig. 3). We found that this network consisted of clear three modules linking
each other. One module (module 1, yellow marked region) was mainly formed of a
set of the DREs in the low MOI condition, and its constituent elements were
many interferon (IFN)-stimulated genes (ISGs): IFIs, MXs, OASs, TRIMs, IFTMs,
IRFs, and STATs. These highly orchestrated webs of various ISGs are induced by
transductions of both IFN signaling and subsequent JAK/STAT signaling (26).
This evidence strongly suggested that module 1 represents the consequences of
activation of both these signaling pathways by the acute antiviral response.
Contrary to module 1, the other two modules (module 2, green marked region;
and module 3, purple marked region) are mainly shaped by a set of the DREs in
the high MOI condition. Module 2 and module 3 were found to comprise fewer
IFN-related genes. While module 2 appeared to be a GAS5-centralized module,
module 3 was composed of chemokines (CXCL1, CXCL2, CXCL3, CX3CL1, and CCL20),
interleukins (IL6, IL1A, IL1B, and IL32), and colony-stimulating factors (CSF2
and CSF3), which are implicated in inflammatory-related cytokine signaling
followed by the acute activation of IFN and JAK/STAT signaling represented in
module 1. In particular, the cluster of module 1 and module 3 probably
represents the transition of the gene regulatory system in response to SARS-
CoV-2 infection. Namely, the cellular system perturbated by SARS-CoV-2
gradually switches to inflammatory signaling (module 3) via IFN and JAK/STAT
signaling (module 1) as the viral load increased. This was consistent with the
clinical observations of COVID-19, and thus may partially explain the process
of cytokine storm syndromes, which is a severe clinical feature of COVID-19
(21, 23). We also performed the same analyses among the four respiratory
viruses, and found that module 3 was exclusive for SARS-CoV-2 (Supplementary
Fig. S1). Collectively, we identified the SARS-CoV-2-perturbated network and
its three modules, which reflected distinctive host cellular functions in
response to SARS-CoV-2 infection.

### Characterization of the SARS-CoV-2-perturbated network at individual
sample level

Figure 4: Sample-specific individual networks around the GAS5-centralized
module. The GAS5-centralized module (module 3) in the SARS-CoV-2-perturbated
network (presented in Fig. 3) is displayed for four representative samples
from each group (mock for SARS-CoV-2-infection (low MOI: 0.2), SARS-
CoV-2-infected (low MOI: 0.2), mock for SARS-CoV-2-infection (high MOI: 2),
and SARS-CoV-2-infected (high MOI: 2)). RCs are represented as edge sizes to
show individual differences. The node size stands for the extent of outdegree.

We next determined how the signaling represented in the SARS-CoV-2-perturbated
network (Fig. 3) changed across samples. To this end, we developed a novel
quantitative method, called relative contribution (RC), to measure the edge
contribution at an individual level. The mathematical definition of RC is
described in Methods. Within a set of pairwise parent-child relations for a
certain child, the RC captures how parent genes influence a child gene in
response to the pairwise parent’s mRNA expression, and it can therefore reveal
local regulatory changes in response to SARS-CoV-2 infection at an individual
sample level. To characterize the individual networks, we calculated RCs for
12 samples within four groups (mock $\times$ 3 for SARS-CoV-2-infected (MOI:
0.2), SARS-CoV-2-infected $\times$ 3 (MOI: 0.2), mock $\times$ 3 for SARS-
CoV-2-infected (MOI: 2), SARS-CoV-2-infected $\times$ 3 (MOI: 2)) involved in
the network generation process in Fig. 3, and selected representative four
samples from each group. By representing RCs as the sizes of edge widths, we
depicted these four sample-specific individual networks (Supplementary
Animation 1), and found that the vicinity of the GAS5-centralized module
(module 2) drastically changed at an RC level (Fig. 4). Interestingly, this
module included GAS5, SNHG8, ZFAS1, SNORD52, SNORD58C, SNORA24, and
LOC100506548, which encode non-coding RNA (ncRNA) genes. Given that GAS5
appears to function as a hub gene, these results suggested that the genes
downstream of GAS5 are regulated by different cellular systems in the mock and
SARS-CoV-2 infections at a local system level. Especially, our results showed
that GAS5, ZFAS1, and SNHG8 were found to be dominant for SLC9B1 in SARS-
CoV-2-infected samples compared with the mock, suggesting the regulatory
system used was significantly different between them (Fig. 4). GAS5 is a
single-strand long ncRNA and one study demonstrated that the mRNA expression
of GAS5 was elevated in response to hepatitis C virus infection and that GAS5
impaired virus replication by the interaction between truncated-GAS5 and HCV
NS3 protein in human cells (27). Combined with this evidence, our results
suggest the possibility that this ncRNA-related module 2 may play a novel
clear role in SARS-CoV-2 infection.

Conversely, of the four individual networks, the two networks for mocks
exhibited no significant change in RC (Fig. 4 and Supplementary Animation 1).
This was consistent with the prerequisite experimental designs as the mock
samples are supposed to exhibit the same behaviour, which further supported
the validity of our method. Moreover, the RC-highlighted edges displaying even
small or no changes explained that their local regulatory system, presented as
a set of pairwise parent-child relationships for one child, did not change
between the individual samples. Collectively, our data have demonstrated that
we can capture the local system differences in network signaling at an
individual level.

### Identification of specific individual networks for patients with COVID-19

Figure 5: The COVID-19-perturbated network analysis. (A) The histograms of
$\Delta$ECv for the biopsy dataset. The X-axis corresponds to the threshold
for the $\Delta$ECv. The Y-axis stands for the number of edges with log scale.
(B) The COVID-19-perturbated network is shown. The network is composed of 127
nodes and 412 edges (including 245 basal edges). The colored solid edges
represent DREs perturbated by COVID-19 (yellow). The dot edges represent the
basal edges (grey). The nodes (green) represent the known drug target genes
(Supplementary Table S1). The node size represents the extent of outdegree.
(C) The top 10 terms of canonical pathway analysis for the genes in the
COVID-19-perturbated network. (D) The Venn diagram shows the numbers of the
$\Delta$ECv-extracted DREs ($\Delta$ECv threshold 2.3) induced by COVID-19
perturbation for biopsy dataset (yellow) overlapped with the two DREs through
the Venn diagram analysis in Fig. 2B.

Finally, we aimed to establish COVID-19 individual networks with a human
biopsy dataset (healthy: two samples; COVID-19 positive: two samples) on the
basis of the estimated basal network model. We expect that the in vivo biopsy
dataset potentially provides a more clinically relevant perspective compared
with the in vitro experiments. Usually, network estimation is impossible with
such a small number of samples owing to the difficulty in acquisition of the
robust network structure, yet our approach using the basal network model was
capable of generating a context-specific network, even with a few samples of a
different dataset (Fig. 1). By using the $B$-spline regression model of the
Bayesian network acquired by the estimation of the basal network, we first
computed the ECv for the preprocessed biopsy dataset, despite the absence of
some genes compared with the dataset used for the basal network estimation. To
obtain DREs, we calculated $\Delta$ECv between healthy (regarded as control)
and COVID-19 samples following the Eq. (2) where $S$ = healthy ($|S|=2$) and
$T$ = COVID-19 ($|T|=2$) (see Methods). The $\Delta$ECvs were distributed over
a broad range, and 4242 DREs were observed at the threshold of 1 for
$\Delta$ECv (Fig. 5A). To extract more reliable DREs induced by COVID-19, we
set the threshold of 2.3 corresponding approximately to $\log_{2}$FC where
FC=5, which resulted in 638 DREs. These DREs were mapped as networks and the
biggest connected component (167 DREs) was depicted with inclusion of the
basal edges, generating the COVID-19-perturbated network, which comprised 127
nodes and 412 edges (Fig. 5B). This network is supposedly a representation of
the distinctive cellular system in patients with COVID-19. The pathway
analysis of genes contained in this network showed that they were involved in
the immune and inflammatory response (Fig. 5C), supporting the consistency of
our established network with the biological observations in COVID-19.

To determine the signatures of the acquired DREs in the COVID-19-perturbated
network, we measured the ECv similarity for a set of the 167 DREs across the
other experimental samples. This result showed that the ECv profiles in
COVID-19 were most similar to the sample of high SARS-CoV-2 viral load in A549
cells in the vitro experiments (Supplementary Fig. S2A), further supporting
that the obtained DREs were associated with SARS-CoV-2 infection. We further
explored the extent to which the COVID-19-related 167 DREs overlapped with the
Venn diagram established in Fig. 2B. We observed that a moderate number of the
DREs were shared by the cell models of SARS-CoV-2 perturbation (Fig. 5D), and
then these overlapped edges were mapped onto the COVID-19-perturbated network
(Supplementary Fig. S2B). Unlike the network observations in Fig. 3, we found
that both the ISG-related webs (module 1) and subsequent cytokine signaling
(module 3) involved in inflammatory cascades were coincidently present in the
COVID-19-perturbated network, indicating that these two modules were continued
to be mutually activated in COVID-19.

Figure 6: Establishment of the COVID-19 patient-specific individual network.
(A) The COVID-19 patient-specific network with RCs represented by edge sizes.
The network comprises 127 nodes and 412 edges (including 245 basal edges). The
colored solid edges represent DREs; SARS-CoV-2 (high MOI: 2) $\cap$
COVID-19-perturbated (magenta), SARS-CoV-2 (lowMOI: 0.2) $\cap$
COVID-19-perturbated (blue), SARS-CoV-2 (high MOI: 2) $\cap$ SARS-CoV-2 (low
MOI: 0.2) $\cap$ COVID-19-perturbated (purple), COVID-19-perturbated exclusive
edges (yellow). The dot edges represent the basal edges (grey). The nodes
(green) represents the known drug target genes (Supplementary Table S1). The
node size stands for the extent of outdegree. (B) Zoomed regions indicated in
Fig. 6A for three individuals (healthy 1, healthy 2, and the patient with
COVID-19). RCs are represented as edge sizes to show individual differences.

To uncover the differences in the local regulatory system, we next examined
the profiles of the COVID-19-perturbated network at an individual level using
the RC method (Fig. 1). As the two COVID-19 samples were originally derived
from a single patient who tested positive for COVID-19, we calculated RCs for
three individuals (healthy 1, healthy 2, and COVID-19 patient). The depiction
of the RC as the edge sizes eventually led to the establishment of the
COVID-19 patient-specific network, which is likely to show how the cellular
system changed in the patients with COVID-19 compared with the healthy
controls (Fig. 6A). The panel of three individual networks dramatically
exhibited a great magnitude of differences, showing that the cellular
regulatory systems were quite distinctive among individuals (Supplementary
Animation 2). In comparison with the SARS-CoV-2-perturbated network
established by the well-organized in vitro experiments using cell lines
(Supplementary Animation 1), this broad range of RC fluctuation for each in
vitro sample probably reflects further differences among individuals. The
representative regions where local regulatory systems are different among
individuals were illustrated in Fig. 6B. In the zoom 1 region, PELI1 is a
parent gene for both TNF and RGS16; these two signals were dominant in the
healthy individuals, but not in the patient with COVID-19. In contrast, the
zoom 2 and 3 regions showed that local signals were clearly different, not
only between the healthy patients and the patient with COVID-19, but also even
between two healthy individuals.

## Discussion

Here, we have presented the host cellular gene networks perturbated by SARS-
CoV-2 both in vitro and in vivo by using our proposed framework for gene
network analysis. As the networks we established to be associated with SARS-
CoV-2 were generated through RNA-seq data, these networks explain how genes
are systematically regulated at the transcriptome level. Although our approach
depends on the initial network estimation with an experimental dataset and may
therefore risk the inclusion of false relationships or the exclusion of true
relationships, we have succeeded in capturing the biologically explainable
immune response systems in human cells induced by SARS-CoV-2 at the level of
signaling networks.

Sensing viruses cause an immune defense system in host cells, and this induces
acute IFN signaling activation followed by expression of IFNs. These IFNs
amplify JAK/STAT signaling to promote the expression of various ISGs and
accelerate subsequent cytokine signaling (26). As illustrated in Fig. 3, the
mutually interacting module of ISGs (module 1) followed by IFN and JAK/STAT
signaling was shown to be an early response to SARS-CoV-2 infection. During
the process of cells exposed to high SARS-CoV-2 viral loads, the signaling
appears to move to the next stage, represented by inflammatory signaling,
including the involvement of various cytokines (Fig. 3). The recent reported
drug, dexamethasone, could be effective for severe patients with COVID-19 by
suppressing these orchestrated inflammatory signaling cascades (28). In this
network, IL6 was located as a hub gene to regulate downstream cascades,
including chemokines and colony-stimulating factors, which are reported to be
increased in patients with COVID-19 (21). The web of chemokines, such as
CXCL1, CXCL2, and CXCL3, may represent how SARS-CoV-2-infected cells present a
signal to induce leukocyte chemotaxis and infiltration. The localization of
ICAM1 in the vicinity of IL6 and chemokines is supportive of this, as ICAM1 is
known to be a scaffold for the accumulation of leukocytes at inflammatory
sites and its expression is regulated by cytokines, including IL6 (29, 30).
This tendency was also observed in the network comparison analyses across four
respiratory viruses, including SARS-CoV-2 (Supplementary Fig. S1C). These data
showed that IL6 was not exclusive to SARS-CoV-2, but a universal factor in
response to respiratory viral infection, except for influenza A virus. Given
that several studies have reported that tocilizumab, an inhibitor of the IL6
receptor, is a potential drug able to suppress the cytokine storm observed in
many critical patients with COVID-19 (31, 32), the accumulated evidence
strongly suggested that IL6 would be a central regulator of the inflammatory
cascade, even from a network perspective. In addition, our network showed that
CSF2 is regulated via various factors, including IL6, which strengthens
previous reports suggesting that CSF2 may be a promising therapeutic target in
combination with IL6 (33, 34). Several recent studies have shown that ACE2
plays a key role in the process of SARS-CoV-2 infection. SARS-CoV-2 enters
into host cells via ACE2 (35), and ACE2 was found to be an ISG in human airway
epithelial cells (36). Considering that the SARS-CoV-2-perturbated network
includes several ISGs (Fig. 3), it can be reasoned that some clues regarding
ACE2 may be present in this network. In this context, we found that ACE2 was
closely located to this network and was downstream of TNFRSF9, ATF3, and
ARRDC3 via ACHE (data not shown); these are potential candidates for further
investigation of the relationship between ACE2 and ISGs. Thus, our networks
provide promising information to elucidate SARS-CoV-2 profiles from a broad
biological perspective.

Our second noteworthy outcome in this study was that we succeeded in the
characterization of sample-specific individual networks by introducing the new
edge-quantitative technique of RC. In particular, although it is impossible to
estimate a network with a small number of samples, such as the four biopsy
samples used in our case, the basal network model that was already obtained
through the analysis of the in vitro dataset with both RC and ECv methods has
led to establishment of the COVID-19 patient-specific individual network. This
process represents how we extrapolate between in vitro and in vivo
experiments. Different samples each exhibit a unique regulatory profile,
especially in actual individuals (such as those obtained from biopsy) rather
than well-controlled in vitro samples (Supplementary Animation 1 and 2). These
results probably reflect a more realistic clinical situation and increase the
importance of making the most of effective utilization of a biopsy dataset. In
the current outbreak of COVID-19, we need to look into both biological and
clinical aspects for exploring COVID-19 therapy. The individual networks
regarding COVID-19 shows the extent to which individuals possess their own
network, which ultimately links to the necessity of personalized treatment.
Therefore, our efforts are a potential contribution to the emerging field of
personalized medicine. The biopsy dataset that we used was not sufficient to
allow interpretation of the comprehensive information through individual
networks in patients with COVID-19, as it contained fewer COVID-19 samples.
More clinical samples from patients with COVID-19 can lead to the
determination of key regulatory systems at a clinical level. We hope that our
panel of network analyses will be of help to the SARS-CoV-2 research field and
to establishment further treatments for COVID-19.

## Methods

### Global gene network estimation and core network extraction

In general, methods for gene network analysis are intended for the extraction
of gene-to-gene regulatory relationships universally underlying given
transcriptome datasets. Unlike commonly existing gene networks, we recently
developed a method to extract sample-specific gene networks. Our method first
estimates a global gene network, called the basal network, that included all
the genes in a dataset using a Bayesian network with $B$-spline nonparametric
regression (37, 24). Bayesian network estimation is capable of capturing
global cause-and-effect relationships among gene expression, rather than
extracting locally co-regulated genes, such as co-expression correlation
networks. This is realized by finding the conditional independencies among
variables. In gene network analysis using a Bayesian network, gene expression
is regarded as an observed sample from the random variables that correspond to
genes or transcriptomes in a cell.

Let $X_{1},\ldots,X_{p}$ be $p$ variables of genes. In a Bayesian network, we
consider the joint density of $p$ variables and assume that it is decomposed
as the product of local conditional densities, such that

$f(X_{1},\ldots,X_{p};\theta_{G})=\prod_{j=1}^{p}f(X_{j}|X_{j_{1}},\ldots,X_{j_{q_{j}}};\theta_{j}),$

where $j_{1},\dots,j_{q_{j}}$ are indices of $q_{j}$ dependent variables of
the $j$-th variable. This decomposition can be represented as a directed
acyclic graph (network) and variables $X_{j_{1}},\ldots,X_{j_{q_{j}}}$ are
connected as parents or inputs of the $j$-th variable in the network.

The $B$-spline nonparametric regression version of the Bayesian network models
gene-to-gene expression relationships as mathematical equations using
$B$-spline curves, such that

$x_{j}=m^{(j)}_{1}(x_{j_{1}})+\cdots+m^{(j)}_{q_{j}}(x_{j_{q_{j}}})+\varepsilon_{j},$
(1)

where $x_{j}$ represents an expression value of the $j$-th gene,
$\varepsilon_{j}$ is the error term normally distributed with mean $0$ and
variance $\sigma_{j}$, and
$m^{(j)}_{k}(x)=\sum_{m=1}^{M}\gamma_{jkm}b_{jkm}(x)$ is a regression function
using $M$ $B$-spline basis functions $b_{jkm}(\cdot)$, and their coefficients
$\gamma_{jkm}$.

The structure search of the Bayesian network corresponds to finding the
decomposition of the joint density. This is implemented by the maximization of
the posterior probability, such that

$p(G|X)\propto\pi(G)\int\prod_{i=1}^{n}f(x_{i1},\cdots,x_{ip};\theta_{G})\pi(\theta_{G}|\lambda)d\theta_{G},$

where $X$ is an $n$-by-$p$ input matrix whose element $x_{ij}$ corresponds to
an expression value of the $i$-th sample for the $j$-th gene, $G$ represents
the network structure, $\pi(G)$ is the prior probability of $G$, $\theta_{G}$
is the parameter vector of the local conditional densities,
$\pi(\theta_{G}|\lambda)$ is the prior distribution of $\theta_{G}$, and
$\lambda$ is the hyperparameter vector. The difficulty in gene network
estimation by the Bayesian network is the step of structure learning for large
networks, because this is known to be an NP-hard problem; namely, an
exponential increase in search space to the number of variables. We have used
the neighbor node sampling and repeat algorithm that realizes the estimation
of the large Bayesian network structure (24). It repeats the subnetwork
estimation many times in parallel for the sampled variable sets by random
walking, and thus it can estimate the large network within a realistic time.

After the basal network estimation, we then quantified every single edge with
respect to a certain sample in terms of the system-level usage of the edge
with the estimated mathematical model. Tanaka et al. (14) defined an edge
contribution value (ECv) of edge $j_{k}\rightarrow j$ as

$ECv_{(u)}(j_{k}\rightarrow j)=m^{(j)}_{k}(x^{(u)}_{j_{k}})$

where $x^{(u)}_{j_{k}}$ represented the expression value of the $j_{k}$-th
gene at a certain sample denoted by $u$, and $m^{(j)}_{k}(\cdot)$ was a
regression function defined in Eq. (1). Note that sample $u$ did not
necessarily have to be a single sample for use in the network estimation. They
proved that ECv can be used for the quantification of edge $j_{k}\rightarrow
j$ with respect to a given sample. To extract sample-specific networks, they
considered the differences of ECvs between two different conditions of
samples, similar to extracting differentially expressed genes by comparing
control and perturbated expressions. They defined $\Delta$ECv as

$\begin{split}&\Delta ECv_{(S,T)}(j_{k}\rightarrow j)=\\\
&\left|\frac{1}{|S|}\sum_{s\in S}ECv_{(s)}(j_{k}\rightarrow
j)-\frac{1}{|T|}\sum_{t\in T}ECv_{(t)}(j_{k}\rightarrow j)\right|,\end{split}$
(2)

where $S$ and $T$ are sets of samples observed in the particular conditions,
respectively, in which $|S|\geq 1$ and $|T|\geq 1$. Note that in the case of
$|S|=|T|=1$, this allows consideration of the differences between just two
samples, e.g. control and perturbated samples. In general, we assume multiple
replicated samples or a set of individual samples for both $S$ and $T$. By
extracting edges and their connected nodes with $\Delta$ECvs greater or equal
to a certain threshold, we can define the sample- or condition-specific core
network from the basal network. In this study, $S$ and $T$ were sets of
infected and control (mock) replicated samples, respectively. As the target
dataset includes control samples for a particular series of experiments, we
can extract certain core networks from them by calculating $\Delta$ECv for the
series of experiments using their corresponding control samples. For example,
we extracted a SARS-CoV-2-perturbated core network by calculating $\Delta$ECvs
for SARS-CoV-2-infected and their corresponding mock-triplicate samples. As
performed in the previous study, we generally employed $\Delta\textrm{ECv}\geq
1.0$ for the threshold of the core network extraction. This approximately
corresponds to 2-fold changes in differentially expressed genes for extracted
genes. Thus, we considered the extracted networks, including edges and nodes,
which showed the significant activation of regulatory systems by the
infection.

### Proposed relative contribution of edges for characterizing individual
networks

This ECv development allowed a new solution for gene network analyses. In the
previous study (14), we succeeded in characterizing network profiles by
calculating ECvs for edges in a $\Delta$ECv-extracted core network with
respect to many samples from patients with cancer. The conventional clustering
onto these calculated ECvs led to the identification of prognosis-related
subgroups. Thus, we demonstrated that the differences and similarities in edge
profiles of the network could be captured as patterns of ECvs. Despite the
high availability of ECvs, it is still impossible to directly compare ECvs
between individual samples, because ECvs have different sizes depending on the
estimated pairwise edge and the sample. The normalization of ECvs across
samples is inappropriate for our purpose owing to the mutual dependency of the
individual network on each sample. Thus, it has not been possible to highlight
the differences in regulatory systems at an individual level.

For these reasons, ECvs are not appropriate to the analysis of individual
networks. To overcome the drawbacks, we have proposed a novel method, relative
contribution (RC), to quantify edges with respect to individual samples using
the estimated gene network model. We hypothesized that the differences of
individual samples in terms of the cellular system can be observed as the
differences in ratios of the contributions of edges connecting to a certain
node in the network. Edges with different samples need to be described as
differently weighted edges according to ratios of effects between parents that
regulate or are connected to a certain gene. In addition, the quantification
of a network with a single sample needs to be independent from other samples
and their distributions. To realize this, we define the relative contribution
of an edge with respect to a sample as

$RC_{(u)}(j_{k}\rightarrow j)=\frac{|ECv_{(u)}(j_{k}\rightarrow
j)|}{\max_{1\leq k^{\prime}\leq q_{j}}|ECv_{(u)}(j_{k^{\prime}}\rightarrow
j)|},$

where $u$ represents a certain sample ($0<RC\leq 1$). That is, an RC of the
edge is a relative strength of the contribution of the edge to the maximum
strength among the parents connecting to the same child node. The reason why
an RC is not divided by the sum of the ECvs is that the range of RCs does not
shrink depending on the number of parents of the child node. A drawback of RCs
is that, if the ratio of ECvs of the parents is not changed, the changes of
parent values do not affect the RCs. However, RCs of their downstreams will be
affected by such changes. Therefore, this drawback is not problematic in terms
of the specification of differences in individual networks. Note that, similar
to ECvs, sample $u$ does not necessarily need to be a single sample used for
the network estimation. As illustrated in Results, we have shown that RCs can
be used to analyze individual networks, even if we have a single sample, or
only a few samples, of gene expression data, as long as a basal network can be
estimated from other datasets. RC therefore offers a significant enhancement
to our framework for gene network analysis. Our data have demonstrated that
the framework, through an integration of the three key pieces – Bayesian
network estimation, ECv, and RC – provides a powerful data-driven solution to
seek biological phenomena through cellular systems ranging from a global level
to an individual level. Our proposed framework is mathematically illustrated
in Supplementary Fig. S3.

### Dataset

The transcriptome dataset GSE147507 was downloaded from NCBI Gene Expression
Omnibus (7). The samples were infected with respiratory viruses, including
SARS-CoV-2, and biological replicates were performed. We first selected
samples exclusive for human RNA-seq with 78 samples. Among the samples, four
samples of the in vivo experiment (biopsy) data were pre-eliminated. The
$\log_{2}$-transformed dataset was filtered to remove genes with a mean
percentile lower than 30$\%$, resulting in 74 samples and 15,258 genes. This
preprocessed dataset of the 74 $\times$ 15258 matrix was used as input for the
basal network estimation. The biopsy dataset eliminated above prior to global
network estimation consisted of four samples (two healthy samples and two
COVID-19-positive samples). The RPM (reads per million) normalized biopsy
dataset was $\log_{2}$-transformed and genes with at least one zero value were
removed to obtain more reliable data. The two technical replicate samples for
COVID-19 were averaged for RC calculation. Following this preprocessing, the
input dataset for RC calculation finally comprised a 3 $\times$ 4516 matrix.
The RNA-seq samples used for $\Delta$ECv calculations in this study were:
SARS-CoV-2 in A549 cells (MOI of 0.2/2 for 24 hr, n=3) and the corresponding
mock (n=3); SARS-CoV-2 in normal human bronchial epithelial (NHBE) cells (MOI
of 2 for 24 hr, n=3) and the corresponding mock (n=3); SARS-CoV-2 in Calu-3
cells (MOI of 2 for 24 hr, n=3) and the corresponding mock (n=3); human
respiratory syncytial virus (RSV) in A549 cells (MOI of 2 for 24 hr, n=3) and
the corresponding mock (n=3); human parainfluenza virus 3 (HPIV3) in A549
cells (MOI of 2 for 24 hr, n=3) and the corresponding mock (n=3); influenza A
virus (IAV) in A549 cells (MOI of 5 for 9 hr, n=2) and the corresponding mock
(n=2); COVID-19 (n=2) and healthy (n=2).

### Pathway analysis

The canonical pathway analysis was performed through the use of Ingenuity
Pathway Analysis software (38).

### Network analysis and visualization

The network visualization and the network analysis were performed using
Cytoscape (version 3.7.2 and 3.8.0) (39). The genes for known drug targets
were acquired from IPA knowledge database (38) and the representative drugs
were listed in Supplementary Table S1.

### Computer environments

All the computation for the network estimation and the ECv calculations in
this study were performed by the SHIROKANE supercomputer system (Shirokane5)
at Human Genome Center, the Institute of Medical Science, the University of
Tokyo, where the computation nodes were equipped with dual Intel Xeon Gold
6154 3.0GHz CPUs and 192GB memory per node.

### Data availability

All the network files generated in this study are provided on the
supplementary data (http://ytlab.jp/suppl/tanaka_arxiv2020/index.html). The
program for network estimation is freely available for SHIROKANE users. The
ECv/RC calculation program is available for non-commercial academic users upon
request.

###### Acknowledgements.

We thank M. Ikeguchi for technical support. The super-computing resource was
provided by Human Genome Center, the Institute of Medical Science, the
University of Tokyo. This work was supported by RIKEN Junior Research
Associate Program. This study was performed using funds provided by PRISM
(Public/Private R$\&$D Investment Strategic Expansion PrograM) in Japan. This
paper format was generated through self-modification of the original template
designed by Ricardo Henriques.

Y.T. (Yoshihisa Tanaka), Y.T.* (Yoshinori Tamada), and Y.O. conceived the
experiments, Y.T., K.H., and M.A.N. conducted the experiments, K.H. visualized
the networks, Y.T., Y.T.*, and Y.O. analyzed the results, Y.T. and Y.T.* wrote
the manuscript, M.A.N., F.Y., and Y.O. reviewed and edited the manuscript.

Y. Tamada and Y. Okuno have a patent application on the individual profiling
and the method for extraction of a sample-specific network used in this study
through the technology licensing organization in Kyoto University. Other
authors declare no conflict of interest.

## Bibliography

## References

  * Zhu et al. (2020) Na Zhu, Dingyu Zhang, Wenling Wang, Xingwang Li, Bo Yang, Jingdong Song, Xiang Zhao, Baoying Huang, Weifeng Shi, Roujian Lu, Peihua Niu, Faxian Zhan, Xuejun Ma, Dayan Wang, Wenbo Xu, Guizhen Wu, George F Gao, Wenjie Tan, and China Novel Coronavirus Investigating and Research Team.  A novel coronavirus from patients with pneumonia in china, 2019.  _N. Engl. J. Med._ , 382(8):727–733, February 2020.  ISSN 0028-4793, 1533-4406.  10.1056/NEJMoa2001017.
  * Wu et al. (2020) Fan Wu, Su Zhao, Bin Yu, Yan-Mei Chen, Wen Wang, Zhi-Gang Song, Yi Hu, Zhao-Wu Tao, Jun-Hua Tian, Yuan-Yuan Pei, Ming-Li Yuan, Yu-Ling Zhang, Fa-Hui Dai, Yi Liu, Qi-Min Wang, Jiao-Jiao Zheng, Lin Xu, Edward C Holmes, and Yong-Zhen Zhang.  A new coronavirus associated with human respiratory disease in china.  _Nature_ , 579(7798):265–269, March 2020.  ISSN 0028-0836, 1476-4687.  10.1038/s41586-020-2008-3.
  * Dong et al. (2020) Ensheng Dong, Hongru Du, and Lauren Gardner.  An interactive web-based dashboard to track COVID-19 in real time.  _Lancet Infect. Dis._ , 20(5):533–534, May 2020\.  ISSN 1473-3099, 1474-4457.  10.1016/S1473-3099(20)30120-1.
  * Hellewell et al. (2020) Joel Hellewell, Sam Abbott, Amy Gimma, Nikos I Bosse, Christopher I Jarvis, Timothy W Russell, James D Munday, Adam J Kucharski, W John Edmunds, Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Sebastian Funk, and Rosalind M Eggo.  Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts.  _Lancet Glob Health_ , 8(4):e488–e496, April 2020\.  ISSN 2214-109X.  10.1016/S2214-109X(20)30074-7.
  * Gordon et al. (2020) David E Gordon, Gwendolyn M Jang, Mehdi Bouhaddou, Jiewei Xu, Kirsten Obernier, Kris M White, Matthew J O’Meara, Veronica V Rezelj, Jeffrey Z Guo, Danielle L Swaney, Tia A Tummino, Ruth Hüttenhain, Robyn M Kaake, Alicia L Richards, Beril Tutuncuoglu, Helene Foussard, Jyoti Batra, Kelsey Haas, Maya Modak, Minkyu Kim, Paige Haas, Benjamin J Polacco, Hannes Braberg, Jacqueline M Fabius, Manon Eckhardt, Margaret Soucheray, Melanie J Bennett, Merve Cakir, Michael J McGregor, Qiongyu Li, Bjoern Meyer, Ferdinand Roesch, Thomas Vallet, Alice Mac Kain, Lisa Miorin, Elena Moreno, Zun Zar Chi Naing, Yuan Zhou, Shiming Peng, Ying Shi, Ziyang Zhang, Wenqi Shen, Ilsa T Kirby, James E Melnyk, John S Chorba, Kevin Lou, Shizhong A Dai, Inigo Barrio-Hernandez, Danish Memon, Claudia Hernandez-Armenta, Jiankun Lyu, Christopher J P Mathy, Tina Perica, Kala Bharath Pilla, Sai J Ganesan, Daniel J Saltzberg, Ramachandran Rakesh, Xi Liu, Sara B Rosenthal, Lorenzo Calviello, Srivats Venkataramanan, Jose Liboy-Lugo, Yizhu Lin, Xi-Ping Huang, Yongfeng Liu, Stephanie A Wankowicz, Markus Bohn, Maliheh Safari, Fatima S Ugur, Cassandra Koh, Nastaran Sadat Savar, Quang Dinh Tran, Djoshkun Shengjuler, Sabrina J Fletcher, Michael C O’Neal, Yiming Cai, Jason C J Chang, David J Broadhurst, Saker Klippsten, Phillip P Sharp, Nicole A Wenzell, Duygu Kuzuoglu-Ozturk, Hao-Yuan Wang, Raphael Trenker, Janet M Young, Devin A Cavero, Joseph Hiatt, Theodore L Roth, Ujjwal Rathore, Advait Subramanian, Julia Noack, Mathieu Hubert, Robert M Stroud, Alan D Frankel, Oren S Rosenberg, Kliment A Verba, David A Agard, Melanie Ott, Michael Emerman, Natalia Jura, Mark von Zastrow, Eric Verdin, Alan Ashworth, Olivier Schwartz, Christophe d’Enfert, Shaeri Mukherjee, Matt Jacobson, Harmit S Malik, Danica G Fujimori, Trey Ideker, Charles S Craik, Stephen N Floor, James S Fraser, John D Gross, Andrej Sali, Bryan L Roth, Davide Ruggero, Jack Taunton, Tanja Kortemme, Pedro Beltrao, Marco Vignuzzi, Adolfo García-Sastre, Kevan M Shokat, Brian K Shoichet, and Nevan J Krogan.  A SARS-CoV-2 protein interaction map reveals targets for drug repurposing.  _Nature_ , 583(7816):459–468, July 2020.  ISSN 0028-0836, 1476-4687.  10.1038/s41586-020-2286-9.
  * Bojkova et al. (2020) Denisa Bojkova, Kevin Klann, Benjamin Koch, Marek Widera, David Krause, Sandra Ciesek, Jindrich Cinatl, and Christian Münch.  Proteomics of SARS-CoV-2-infected host cells reveals therapy targets.  _Nature_ , 583(7816):469–472, July 2020.  ISSN 0028-0836, 1476-4687.  10.1038/s41586-020-2332-7.
  * Blanco-Melo et al. (2020) Daniel Blanco-Melo, Benjamin E Nilsson-Payant, Wen-Chun Liu, Skyler Uhl, Daisy Hoagland, Rasmus Møller, Tristan X Jordan, Kohei Oishi, Maryline Panis, David Sachs, Taia T Wang, Robert E Schwartz, Jean K Lim, Randy A Albrecht, and Benjamin R tenOever.  Imbalanced host response to SARS-CoV-2 drives development of COVID-19.  _Cell_ , 181(5):1036–1045.e9, May 2020.  ISSN 0092-8674, 1097-4172.  10.1016/j.cell.2020.04.026.
  * Yan et al. (2018) Jingwen Yan, Shannon L Risacher, Li Shen, and Andrew J Saykin.  Network approaches to systems biology analysis of complex disease: integrative methods for multi-omics data.  _Brief. Bioinform._ , 19(6):1370–1381, November 2018.  ISSN 1467-5463, 1477-4054.  10.1093/bib/bbx066.
  * Recanatini and Cabrelle (2020) Maurizio Recanatini and Chiara Cabrelle.  Drug research meets network science: Where are we?  _J. Med. Chem._ , May 2020.  ISSN 0022-2623, 1520-4804.  10.1021/acs.jmedchem.9b01989.
  * Guzzi et al. (2020) Pietro H Guzzi, Daniele Mercatelli, Carmine Ceraolo, and Federico M Giorgi.  Master regulator analysis of the SARS-CoV-2/Human interactome.  _J. Clin. Med. Res._ , 9(4), April 2020.  ISSN 1918-3003, 2077-0383.  10.3390/jcm9040982.
  * Zhou et al. (2020a) Yadi Zhou, Yuan Hou, Jiayu Shen, Yin Huang, William Martin, and Feixiong Cheng.  Network-based drug repurposing for novel coronavirus 2019-nCoV/SARS-CoV-2.  _Cell Discov_ , 6:14, March 2020a.  ISSN 2056-5968.  10.1038/s41421-020-0153-3.
  * Gysi et al. (2020) Deisy Morselli Gysi, Ítalo Do Valle, Marinka Zitnik, Asher Ameli, Xiao Gan, Onur Varol, Helia Sanchez, Rebecca Marlene Baron, Dina Ghiassian, Joseph Loscalzo, and Albert-László Barabási.  Network medicine framework for identifying drug repurposing opportunities for COVID-19.  _ArXiv_ , April 2020.  ISSN 2331-8422.
  * Fagone et al. (2020) Paolo Fagone, Rosella Ciurleo, Salvo Danilo Lombardo, Carmelo Iacobello, Concetta Ilenia Palermo, Yehuda Shoenfeld, Klaus Bendtzen, Placido Bramanti, and Ferdinando Nicoletti.  Transcriptional landscape of SARS-CoV-2 infection dismantles pathogenic pathways activated by the virus, proposes unique sex-specific differences and predicts tailored therapeutic strategies.  _Autoimmun. Rev._ , page 102571, May 2020.  ISSN 1568-9972, 1873-0183.  10.1016/j.autrev.2020.102571.
  * Tanaka et al. (2020) Yoshihisa Tanaka, Yoshinori Tamada, Marie Ikeguchi, Fumiyoshi Yamashita, and Yasushi Okuno.  System-Based differential gene network analysis for characterizing a Sample-Specific subnetwork.  _Biomolecules_ , 10(2), February 2020.  ISSN 2218-273X.  10.3390/biom10020306.
  * Margolin et al. (2006) Adam A Margolin, Ilya Nemenman, Katia Basso, Chris Wiggins, Gustavo Stolovitzky, Riccardo Dalla Favera, and Andrea Califano.  ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context.  _BMC Bioinformatics_ , 7 Suppl 1:S7, March 2006.  ISSN 1471-2105.  10.1186/1471-2105-7-S1-S7.
  * Araki et al. (2009) Hiromitsu Araki, Yoshinori Tamada, Seiya Imoto, Ben Dunmore, Deborah Sanders, Sally Humphrey, Masao Nagasaki, Atsushi Doi, Yukiko Nakanishi, Kaori Yasuda, Yuki Tomiyasu, Kousuke Tashiro, Cristin Print, D Stephen Charnock-Jones, Satoru Kuhara, and Satoru Miyano.  Analysis of PPAR$\alpha$-dependent and PPAR$\alpha$-independent transcript regulation following fenofibrate treatment of human endothelial cells.  _Angiogenesis_ , 12(3):221–229, September 2009\.  ISSN 0969-6970, 1573-7209.  10.1007/s10456-009-9142-8.
  * Wang et al. (2012) Li Wang, Daniel G Hurley, Wendy Watkins, Hiromitsu Araki, Yoshinori Tamada, Anita Muthukaruppan, Louis Ranjard, Eliane Derkac, Seiya Imoto, Satoru Miyano, Edmund J Crampin, and Cristin G Print.  Cell cycle gene networks are associated with melanoma prognosis.  _PLoS One_ , 7(4):e34247, April 2012.  ISSN 1932-6203.  10.1371/journal.pone.0034247.
  * Affara et al. (2013) Muna Affara, Debbie Sanders, Hiromitsu Araki, Yoshinori Tamada, Benjamin J Dunmore, Sally Humphreys, Seiya Imoto, Christopher Savoie, Satoru Miyano, Satoru Kuhara, David Jeffries, Cristin Print, and D Stephen Charnock-Jones.  Vasohibin-1 is identified as a master-regulator of endothelial cell apoptosis using gene network analysis.  _BMC Genomics_ , 14:23, January 2013.  ISSN 1471-2164.  10.1186/1471-2164-14-23.
  * Singh et al. (2018) Arun J Singh, Stephen A Ramsey, Theresa M Filtz, and Chrissa Kioussi.  Differential gene regulatory networks in development and disease.  _Cell. Mol. Life Sci._ , 75(6):1013–1025, March 2018.  ISSN 1420-682X, 1420-9071.  10.1007/s00018-017-2679-6.
  * Yan et al. (2016) Koon-Kiu Yan, Daifeng Wang, Anurag Sethi, Paul Muir, Robert Kitchen, Chao Cheng, and Mark Gerstein.  Cross-Disciplinary network comparison: Matchmaking between hairballs.  _Cell Syst_ , 2(3):147–157, March 2016.  ISSN 2405-4712.  10.1016/j.cels.2016.02.014.
  * Huang et al. (2020) Chaolin Huang, Yeming Wang, Xingwang Li, Lili Ren, Jianping Zhao, Yi Hu, Li Zhang, Guohui Fan, Jiuyang Xu, Xiaoying Gu, Zhenshun Cheng, Ting Yu, Jiaan Xia, Yuan Wei, Wenjuan Wu, Xuelei Xie, Wen Yin, Hui Li, Min Liu, Yan Xiao, Hong Gao, Li Guo, Jungang Xie, Guangfa Wang, Rongmeng Jiang, Zhancheng Gao, Qi Jin, Jianwei Wang, and Bin Cao.  Clinical features of patients infected with 2019 novel coronavirus in wuhan, china.  _Lancet_ , 395(10223):497–506, February 2020\.  ISSN 0140-6736, 1474-547X.  10.1016/S0140-6736(20)30183-5.
  * Guan et al. (2020) Wei-Jie Guan, Zheng-Yi Ni, Yu Hu, Wen-Hua Liang, Chun-Quan Ou, Jian-Xing He, Lei Liu, Hong Shan, Chun-Liang Lei, David S C Hui, Bin Du, Lan-Juan Li, Guang Zeng, Kwok-Yung Yuen, Ru-Chong Chen, Chun-Li Tang, Tao Wang, Ping-Yan Chen, Jie Xiang, Shi-Yue Li, Jin-Lin Wang, Zi-Jing Liang, Yi-Xiang Peng, Li Wei, Yong Liu, Ya-Hua Hu, Peng Peng, Jian-Ming Wang, Ji-Yang Liu, Zhong Chen, Gang Li, Zhi-Jian Zheng, Shao-Qin Qiu, Jie Luo, Chang-Jiang Ye, Shao-Yong Zhu, Nan-Shan Zhong, and China Medical Treatment Expert Group for Covid-19.  Clinical characteristics of coronavirus disease 2019 in china.  _N. Engl. J. Med._ , 382(18):1708–1720, April 2020.  ISSN 0028-4793, 1533-4406.  10.1056/NEJMoa2002032.
  * Mehta et al. (2020) Puja Mehta, Daniel F McAuley, Michael Brown, Emilie Sanchez, Rachel S Tattersall, Jessica J Manson, and HLH Across Speciality Collaboration, UK.  COVID-19: consider cytokine storm syndromes and immunosuppression.  _Lancet_ , 395(10229):1033–1034, March 2020.  ISSN 0140-6736, 1474-547X.  10.1016/S0140-6736(20)30628-0.
  * Tamada et al. (2011) Yoshinori Tamada, Seiya Imoto, Hiromitsu Araki, Masao Nagasaki, Cristin Print, D Stephen Charnock-Jones, and Satoru Miyano.  Estimating genome-wide gene networks using nonparametric bayesian network models on massively parallel computers.  _IEEE/ACM Trans. Comput. Biol. Bioinform._ , 8(3):683–697, May 2011.  ISSN 1545-5963, 1557-9964.  10.1109/TCBB.2010.68.
  * Barabási and Oltvai (2004) Albert-László Barabási and Zoltán N Oltvai.  Network biology: understanding the cell’s functional organization.  _Nat. Rev. Genet._ , 5(2):101–113, February 2004\.  ISSN 1471-0056.  10.1038/nrg1272.
  * Mesev et al. (2019) Emily V Mesev, Robert A LeDesma, and Alexander Ploss.  Decoding type I and III interferon signalling during viral infection.  _Nat Microbiol_ , 4(6):914–924, June 2019.  ISSN 2058-5276.  10.1038/s41564-019-0421-x.
  * Qian et al. (2016) Xijing Qian, Chen Xu, Ping Zhao, and Zhongtian Qi.  Long non-coding RNA GAS5 inhibited hepatitis C virus replication by binding viral NS3 protein.  _Virology_ , 492:155–165, May 2016.  ISSN 0042-6822, 1096-0341.  10.1016/j.virol.2016.02.020.
  * RECOVERY Collaborative Group et al. (2020) RECOVERY Collaborative Group, Peter Horby, Wei Shen Lim, Jonathan R Emberson, Marion Mafham, Jennifer L Bell, Louise Linsell, Natalie Staplin, Christopher Brightling, Andrew Ustianowski, Einas Elmahi, Benjamin Prudon, Christopher Green, Timothy Felton, David Chadwick, Kanchan Rege, Christopher Fegan, Lucy C Chappell, Saul N Faust, Thomas Jaki, Katie Jeffery, Alan Montgomery, Kathryn Rowan, Edmund Juszczak, J Kenneth Baillie, Richard Haynes, and Martin J Landray.  Dexamethasone in hospitalized patients with covid-19 - preliminary report.  _N. Engl. J. Med._ , July 2020.  ISSN 0028-4793, 1533-4406.  10.1056/NEJMoa2021436.
  * Meager (1999) A Meager.  Cytokine regulation of cellular adhesion molecule expression in inflammation.  _Cytokine Growth Factor Rev._ , 10(1):27–39, March 1999.  ISSN 1359-6101.  10.1016/s1359-6101(98)00024-0.
  * Velazquez-Salinas et al. (2019) Lauro Velazquez-Salinas, Antonio Verdugo-Rodriguez, Luis L Rodriguez, and Manuel V Borca.  The role of interleukin 6 during viral infections.  _Front. Microbiol._ , 10:1057, May 2019.  ISSN 1664-302X.  10.3389/fmicb.2019.01057.
  * Fu et al. (2020) Binqing Fu, Xiaoling Xu, and Haiming Wei.  Why tocilizumab could be an effective treatment for severe COVID-19?  _J. Transl. Med._ , 18(1):164, April 2020.  ISSN 1479-5876.  10.1186/s12967-020-02339-3.
  * Xu et al. (2020) Xiaoling Xu, Mingfeng Han, Tiantian Li, Wei Sun, Dongsheng Wang, Binqing Fu, Yonggang Zhou, Xiaohu Zheng, Yun Yang, Xiuyong Li, Xiaohua Zhang, Aijun Pan, and Haiming Wei.  Effective treatment of severe COVID-19 patients with tocilizumab.  _Proc. Natl. Acad. Sci. U. S. A._ , 117(20):10970–10975, May 2020.  ISSN 0027-8424, 1091-6490.  10.1073/pnas.2005615117.
  * Zhou et al. (2020b) Yonggang Zhou, Binqing Fu, Xiaohu Zheng, Dongsheng Wang, Changcheng Zhao, Yingjie Qi, Rui Sun, Zhigang Tian, Xiaoling Xu, and Haiming Wei.  Pathogenic t-cells and inflammatory monocytes incite inflammatory storms in severe COVID-19 patients.  _Natl Sci Rev_ , 7(6):998–1002, June 2020b.  ISSN 2095-5138.  10.1093/nsr/nwaa041.
  * Lang et al. (2020) Frederick M Lang, Kevin M-C Lee, John R Teijaro, Burkhard Becher, and John A Hamilton.  GM-CSF-based treatments in COVID-19: reconciling opposing therapeutic approaches.  _Nat. Rev. Immunol._ , June 2020.  ISSN 1474-1733, 1474-1741.  10.1038/s41577-020-0357-7.
  * Hoffmann et al. (2020) Markus Hoffmann, Hannah Kleine-Weber, Simon Schroeder, Nadine Krüger, Tanja Herrler, Sandra Erichsen, Tobias S Schiergens, Georg Herrler, Nai-Huei Wu, Andreas Nitsche, Marcel A Müller, Christian Drosten, and Stefan Pöhlmann.  SARS-CoV-2 cell entry depends on ACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor.  _Cell_ , 181(2):271–280.e8, April 2020.  ISSN 0092-8674, 1097-4172.  10.1016/j.cell.2020.02.052.
  * Ziegler et al. (2020) Carly G K Ziegler, Samuel J Allon, Sarah K Nyquist, Ian M Mbano, Vincent N Miao, Constantine N Tzouanas, Yuming Cao, Ashraf S Yousif, Julia Bals, Blake M Hauser, Jared Feldman, Christoph Muus, Marc H Wadsworth, 2nd, Samuel W Kazer, Travis K Hughes, Benjamin Doran, G James Gatter, Marko Vukovic, Faith Taliaferro, Benjamin E Mead, Zhiru Guo, Jennifer P Wang, Delphine Gras, Magali Plaisant, Meshal Ansari, Ilias Angelidis, Heiko Adler, Jennifer M S Sucre, Chase J Taylor, Brian Lin, Avinash Waghray, Vanessa Mitsialis, Daniel F Dwyer, Kathleen M Buchheit, Joshua A Boyce, Nora A Barrett, Tanya M Laidlaw, Shaina L Carroll, Lucrezia Colonna, Victor Tkachev, Christopher W Peterson, Alison Yu, Hengqi Betty Zheng, Hannah P Gideon, Caylin G Winchell, Philana Ling Lin, Colin D Bingle, Scott B Snapper, Jonathan A Kropski, Fabian J Theis, Herbert B Schiller, Laure-Emmanuelle Zaragosi, Pascal Barbry, Alasdair Leslie, Hans-Peter Kiem, Joanne L Flynn, Sarah M Fortune, Bonnie Berger, Robert W Finberg, Leslie S Kean, Manuel Garber, Aaron G Schmidt, Daniel Lingwood, Alex K Shalek, Jose Ordovas-Montanes, HCA Lung Biological Network. Electronic address<EMAIL_ADDRESS>and HCA Lung Biological Network.  SARS-CoV-2 receptor ACE2 is an Interferon-Stimulated gene in human airway epithelial cells and is detected in specific cell subsets across tissues.  _Cell_ , 181(5):1016–1035.e19, May 2020.  ISSN 0092-8674, 1097-4172.  10.1016/j.cell.2020.04.035.
  * Imoto et al. (2002) Seiya Imoto, Takao Goto, and Satoru Miyano.  Estimation of genetic networks and functional structures between genes by using bayesian networks and nonparametric regression.  _Pac. Symp. Biocomput._ , pages 175–186, 2002.  ISSN 2335-6936, 2335-6928.
  * Krämer et al. (2014) Andreas Krämer, Jeff Green, Jack Pollard, Jr, and Stuart Tugendreich.  Causal analysis approaches in ingenuity pathway analysis.  _Bioinformatics_ , 30(4):523–530, February 2014\.  ISSN 1367-4803, 1367-4811.  10.1093/bioinformatics/btt703.
  * Shannon et al. (2003) Paul Shannon, Andrew Markiel, Owen Ozier, Nitin S Baliga, Jonathan T Wang, Daniel Ramage, Nada Amin, Benno Schwikowski, and Trey Ideker.  Cytoscape: a software environment for integrated models of biomolecular interaction networks.  _Genome Res._ , 13(11):2498–2504, November 2003\.  ISSN 1088-9051.  10.1101/gr.1239303.

*format=largeformat

Supplementary Fig. S1: Network comparison analyses across four respiratory
viruses. (A) The histograms of $\Delta$ECv for each respiratory virus as
indicated: SARS-CoV-2 (MOI: 2), HPIV3, IAV, and RSV. $\Delta$ECvs were
calculated following the Eq. (2) where $S$ = virus-infected and $T$ =
corresponding mock samples for each virus (see Methods). The X-axis
corresponds to the threshold for each $\Delta$ECv. The Y-axis stands for the
number of edges on a log scale. (B) The Venn diagram represents the numbers of
$\Delta$ECv-extracted edges for all respiratory viruses with a $\Delta$ECv
threshold of 1.0. (C) The respiratory viruses-shared network comprised of 62
nodes and 116 edges (including 53 basal edges). The colored solid edges
represent DREs; SARS-CoV-2 $\cap$ IAV $\cap$ HPIV3 $\cap$ RSV (purple), SARS-
CoV-2 $\cap$ RSV $\cap$ HPIV3 (red), IAV $\cap$ RSV $\cap$ HPIV3 (blue). The
top 10 terms of canonical pathway analysis for the genes of $\Delta$ECv-
extracted DREs shared by at least three viruses in the Venn diagram
(Supplementary Fig. S1B). (D) SARS-CoV-2 specific network comprising 182 nodes
and 295 edges (including 171 basal edges). The solid edges (magenta) represent
DREs for SARS-CoV-2 (MOI: 2). The dot edges represent the basal edges (grey).
The size of the node represents the extent of outdegree. The nodes (green) are
target genes for existing drugs (Supplementary Table S1). The top 10 terms of
canonical pathway analysis for the genes of $\Delta$ECv-extracted DREs
exclusive for SARS-CoV-2 in the Venn diagram (Supplementary Fig. S1B).
Supplementary Fig. S2: Multiple analyses for generating the
COVID-19-perturbated network. (A) The similarity heatmap is shown and samples
for comparisons are labeled as indicated. Similarity is calculated with cosine
distance method for ECvs of the 167 DREs. (B) The COVID-19 patient-specific
network in combination with the Venn diagram analysis (Fig. 5D). The network
is composed of 127 nodes and 412 edges (including 245 basal edges). The
colored solid edges represent DREs; SARS-CoV-2 (high MOI: 2) $\cap$
COVID-19-perturbated (magenta), SARS-CoV-2 (low MOI: 0.2) $\cap$
COVID-19-perturbated (blue), SARS-CoV-2 (high MOI: 2) $\cap$ SARS-CoV-2 (low
MOI: 0.2) $\cap$ COVID-19-perturbated (purple), COVID-19-perturbated exclusive
edges (yellow). The dot edges represent the basal edges (grey). The nodes
(green) represent the known drug target genes (Supplementary Table S1). The
node size represents the extent of outdegree.  Supplementary Fig. S3:
Mathematical illustration of our proposed framework for the gene network
analysis. The centered hairball (blue) represents a basal network. The network
(magenta) is a core network extracted by the $\Delta$ECv calculation.


# The de Rham cohomology
of the algebra of polynomial functions
on a simplicial complex

Igor Baskov

###### Abstract.

We consider the algebra $A^{0}(X)$ of polynomial functions on a simplicial
complex $X$. The algebra $A^{0}(X)$ is the $0$th component of Sullivan’s dg-
algebra $A^{\bullet}(X)$ of polynomial forms on $X$. All algebras are over an
arbitrary field $k$ of characteristic $0$.

Our main interest lies in computing the de Rham cohomology of the algebra
$A^{0}(X)$, that is, the cohomology of the universal dg-algebra
$\Omega^{\bullet}_{A^{0}(X)}$. There is a canonical morphism of dg-algebras
$P:\Omega^{\bullet}_{A^{0}(X)}\to A^{\bullet}(X)$. We prove that $P$ is a
quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^{0}(X)$
is canonically isomorphic to the cohomology of the simplicial complex $X$ with
coefficients in $k$. Moreover, for $k=\mathbb{Q}$ the dg-algebra
$\Omega^{\bullet}_{A^{0}(X)}$ is a model of the simplicial complex $X$ in the
sense of rational homotopy theory. Our result shows that for the algebra
$A^{0}(X)$ the statement of Grothendieck’s comparison theorem holds.

This research was supported by the Ministry of Science and Higher Education of
the Russian Federation, agreement 075-15-2019-1620 date 08/11/2019 and
075-15-2022-289 date 06/04/2022

##  1\. Introduction

All algebras and dg-algebras are commutative over a field $k$ of
characteristic $0$. In [13, Section $7$] Sullivan introduces the dg-algebra
$A^{\bullet}(X)$ of polynomial forms on a simplicial complex $X$. The algebra
$A^{0}(X)$ of the degree zero elements of $A^{\bullet}(X)$ is the algebra of
polynomial functions on $X$. The cohomology of the dg-algebra $A^{\bullet}(X)$
is isomorphic to $H^{\bullet}(X,k)$. One can ask what natural dg-algebras are
weakly equivalent to $A^{\bullet}(X)$. One such candidate is the universal dg-
algebra $\Omega^{\bullet}_{A^{0}(X)}$ on the algebra $A^{0}(X)$ of polynomial
functions on $X$. There is a canonical morphism of dg-algebras
$P:\Omega^{\bullet}_{A^{0}(X)}\to A^{\bullet}(X)$.

The main result is Theorem 5.2, where we prove that $P$ is a quasi-
isomorphism.

In [10] the authors prove that the morphism $P$ is surjective and give a
description of its kernel. In [6] and [12, Appendix G(i)] another description
of the kernel is given. In [7, Example $3.8$], Gómez establishes that the
morphism $P$ is not a quasi-isomorphism, which contradicts our main result,
Theorem 5.2. We were able to correct the erroneous computation of Gómez in
Remark 5.3.

Grothendieck proved that for a smooth $\mathbb{C}$-algebra $A$ the cohomology
groups of the algebraic de Rham complex $\Omega^{\bullet}_{A}$ are isomorphic
to the cohomology groups of the space $\mathop{\mathrm{Spec}}A$ with complex
analytic topology, see [9, Theorem $1$]. The algebra $A^{0}(X)$ is not smooth
in general and the result of Grothendieck does not hold for general algebras,
see [1, Example $4.4$].

The result of this paper can be used in order to give another proof of the
similar result for the algebra of piecewise polynomial functions on a
polyhedron, which is known due to [2, Theorem $51$].

### Acknowledgement

I would like to thank Dr. Semën Podkorytov for his immense help, fruitful
discussions and the idea to consider Čech resolution in order to prove the
result. I am grateful to the St. Petersburg Department of Steklov Mathematical
Institute of Russian Academy of Sciences for their financial assistance.

##  2\. Simplicial complexes

######  Definition 2.1.

We call a set $X$ of finite non-empty subsets of a finite set $E$ a simplicial
complex if for every $v\in E$ we have $\\{v\\}\in X$ and for every $s\in X$
and every non-empty subset $s^{\prime}\subset s$ we have $s^{\prime}\in X$. We
denote by $V(X)$ the set $E$ and call its elements vertices of $X$. The sets
$s\in X$ of cardinality $m+1$ are called $m$-simplices. A simplicial complex
$Y$ is a subcomplex of $X$ if for every $s\in Y$ we have $s\in X$.

We denote by $T_{p}(X)$ the set of all sequences $u=(u_{0},\dots,u_{p})$ of
vertices of $X$ for $p\geq-1$. We denote by $\partial_{i}u$ the sequence
$(u_{0},\dots,\widehat{u_{i}},\dots,u_{p})$. For a vertex $v$ we denote by
$v*u$ the sequence $(v,u_{0},\dots,u_{p})$. The symmetric group $\Sigma_{p+1}$
acts on $T_{p}(X)$.

Consider a sequence of vertices $u\in T_{p}(X)$. We denote by $\mathrm{St}\,u$
the star of $u$, that is the smallest subcomplex of $X$ containing all the
simplices containing the vertices $u_{i}$. If $p=-1$, we have
$\mathrm{St}\,u=X$. If the sequence $u$ spans a simplex in $X$, then
$\mathrm{St}\,u$ is the star of this simplex. If $p\geq 0$ and the sequence
$u$ does not span a simplex, we have $\mathrm{St}\,u=\varnothing$. For a
subcomplex $Y$ of $X$ we denote by $\mathrm{St}\,_{Y}u$ the smallest
subcomplex of $Y$ containing all the simplices containing the vertices
$u_{i}$. If $u\notin T_{p}(Y)$ then $\mathrm{St}\,_{Y}u=\varnothing$.

##  3\. Sullivan’s dg-algebra of polynomial forms

For a simplicial complex $X$ we define the dg-algebra $A^{\bullet}(X)$
following Sullivan, see [13, Section $7$], [8]. For an $m$-simplex $a$
consider the dg-algebra

$A^{\bullet}(a):=\dfrac{\Lambda\left(t_{v},dt_{v}\mid\mathrm{deg}(t_{v})=0,\,\mathrm{deg}(dt_{v})=1,\,v\in
a\right)}{\left(\sum_{v\in a}t_{v}-1,\sum_{v\in a}dt_{v}\right)}$

with the differential $t_{v}\mapsto dt_{v}$ for $v\in a$.

For a simplex $b$ such that $b\subset a$ one has a natural morphism of dg-
algebras

$|_{b}:A^{\bullet}(a)\to A^{\bullet}(b),\qquad
t_{v}\mapsto\begin{cases}0,\,v\notin b,\\\ t_{v},\,v\in b.\end{cases}$

Then an element $\omega=(\omega_{a})_{a\in X}$ of $A^{\bullet}(X)$ is a
collection of elements $\omega_{a}\in A^{\bullet}(a)$ such that for two
simplices $b\subset a$ one has $\omega_{a}|_{b}=\omega_{b}$.

We call the algebra $A^{0}(X)$ the algebra of polynomial functions on $X$.
This algebra has another description as a quotient of a Stanley-Reisner
algebra, see [3].

An inclusion of simplicial complexes $Y\subset X$ gives rise to the
restriction morphism of dg-algebras

${}|_{Y}:A^{\bullet}(X)\to A^{\bullet}(Y).$

######  Lemma 3.1.

The above restriction morphism is surjective.

###### Proof.

This fact is quite nontrivial, see [13, Section $7$].

∎

We introduce the double graded vector space $\mathcal{D}^{p,q}$,
$p,q\in\mathbb{Z}$ as follows. For $p\leq-2$ set $\mathcal{D}^{p,q}=0$ and for
$p\geq-1$, we define $\mathcal{D}^{p,q}$ as the subspace of

$\prod_{u\in T_{p}(X)}A^{q}(\mathrm{St}\,u)$

consisting of families of forms $\omega_{u}\in A^{q}(\mathrm{St}\,u)$, such
that for any $\sigma\in\Sigma_{p+1}$ we have

$\omega_{\sigma u}=(\mathrm{sgn}\mathop{\sigma})\omega_{u}.$

We define the linear map

$\delta:\mathcal{D}^{p,q}\to\mathcal{D}^{p+1,q}.$

For $p\geq-1$ and for

$\omega=(\omega_{u})_{u\in T_{p}(X)}\in\mathcal{D}^{p,q}$

we set the value of $\delta\omega$ on $s\in T_{p+1}(X)$ as

$(\delta\omega)_{s}=\sum_{i=0}^{p+1}(-1)^{i}\omega_{\partial_{i}s}|_{\mathrm{St}\,s}.$

The differential $d$ on $A^{\bullet}(\mathrm{St}\,u)$ gives rise to a
differential $d$ on $\mathcal{D}^{p,\bullet}$ for each $p$.

######  Proposition 3.2.

The map $\delta$ is a differential on $\mathcal{D}^{\bullet,q}$ for each
$q\in\mathbb{Z}$. Moreover, the double graded vector space
$\mathcal{D}^{\bullet,\bullet}$ together with $\delta$ and $d$ forms a double
complex in the sense that $d\,\delta=\delta\,d$.

######  Proposition 3.3.

The complex

${0}$${\mathcal{D}^{-1,q}}$${\mathcal{D}^{0,q}}$${\mathcal{D}^{1,q}}$${\dots}$${A^{q}(X)}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

is exact.

###### Proof.

The proof is similar to that of Proposition 4.6 below and relies on Lemma 3.1.
In this case one can use the partition of unity $t_{v},\,v\in V(X)$, instead
of $\rho_{v},v\in V(X)$. ∎

##  4\. The dg-algebra of de Rham forms

To a $k$-algebra $A$ one associates the commutative dg-algebra
$\Omega^{\bullet}_{A}$ ([11, Theorem $3.2$]) with $\Omega^{0}_{A}=A$. It has
the following universal property: for any dg-algebra $E^{\bullet}$ and any
algebra homomorphism $f:A\to E^{0}$ there exists a unique morphism of dg-
algebras $F:\Omega^{\bullet}_{A}\to E^{\bullet}$ such that $F|_{A}=f$:

${A}$${\Omega^{\bullet}_{A}}$${E^{\bullet}.}$$\scriptstyle{f}$$\scriptstyle{F}$

The elements of $\Omega^{q}_{A}$ are called algebraic $q$-forms. The dg-
algebra $\Omega^{\bullet}_{A}$ is covariant in the algebra $A$. We will simply
write $\Omega^{\bullet}(X)$ for the dg-algebra $\Omega^{\bullet}_{A^{0}(X)}$.

Inclusion of simplicial complexes $Y\subset X$ gives rise to the restriction
morphism of dg-algebras

${}|_{Y}:\Omega^{\bullet}(X)\to\Omega^{\bullet}(Y).$

######  Lemma 4.1.

Suppose $A$ and $B$ are $k$-algebras and $\varphi:A\to B$ is a surjective
homomorphism of algebras. Then the induced morphism
$\Omega_{\varphi}:\Omega^{\bullet}_{A}\to\Omega^{\bullet}_{B}$ is surjective
and its kernel is the ideal of $\Omega^{\bullet}_{A}$ generated by
$\mathrm{Ker}\varphi$ and $d(\mathrm{Ker}\varphi)$.

From this and Lemma 3.1 it follows that for an inclusion of simplicial
complexes $Y\subset X$ the restriction morphism
${}|_{Y}:\Omega^{\bullet}(X)\to\Omega^{\bullet}(Y)$ is surjective.

###### Proof.

See [2, Lemma $6$]. ∎

Let us introduce the following elements $t_{v}$ of $A^{0}(X)$. For a vertex
$v\in V(X)$ and an $m$-simplex $a\in X$ set $(t_{v})_{a}=0$ if $v\notin a$ and
$(t_{v})_{a}=t_{v}\in A^{0}(a)$ if $v\in a$.

######  Lemma 4.2.

Take a simplicial complex $X$ and a subcomplex $Y\subset X$. Suppose
$\omega\in\Omega^{q}(Y)$ is such that $\omega|_{\mathrm{St}\,_{Y}(v)}=0$ for a
vertex $v\in V(X)$. Then $t_{v}^{2}|_{Y}\,\omega=0$.

###### Proof.

First, if $v\notin V(Y)$, then $t_{v}|_{Y}=0$ and the claim follows.

Assume $v\in V(Y)$. By Lemma 3.1 and Lemma 4.1, the form $\omega$ lies in the
dg-ideal $I$ of $\Omega^{\bullet}(X)$ generated by the elements $m\in
A^{0}(X)$ with the restriction to $\mathrm{St}\,_{Y}(v)$ being zero, therefore
$t_{v}|_{Y}\,m=0$. It is enough to consider the cases $\omega=m$ and
$\omega=dm$. We have $t_{v}^{2}|_{Y}\,m=0$ and
$t_{v}^{2}|_{Y}\,dm=t_{v}|_{Y}\,d(t_{v}|_{Y}\,m)-t_{v}|_{Y}\,mdt_{v}|_{Y}=0$.
∎

We introduce the double graded vector space $\mathcal{C}^{p,q}$,
$p,q\in\mathbb{Z}$ as follows. For $p\leq-2$ set $\mathcal{C}^{p,q}=0$ and for
$p\geq-1$, we define $\mathcal{C}^{p,q}$ as the subspace of

$\prod_{u\in T_{p}(X)}\Omega^{q}(\mathrm{St}\,u)$

consisting of families of forms $\omega_{u}\in\Omega^{q}(\mathrm{St}\,u)$,
such that for any $\sigma\in\Sigma_{p+1}$ we have

$\omega_{\sigma u}=(\mathrm{sgn}\mathop{\sigma})\omega_{u}.$

We define the linear map

$\delta:\mathcal{C}^{p,q}\to\mathcal{C}^{p+1,q}.$

For $p\geq-1$ and for

$\omega=(\omega_{u})_{u\in T_{p}(X)}\in\mathcal{C}^{p,q}$

we set the value of $\delta\omega$ on $s\in T_{p+1}(X)$ as

$(\delta\omega)_{s}=\sum_{i=0}^{p+1}(-1)^{i}\omega_{\partial_{i}s}|_{\mathrm{St}\,s}.$

The differential $d$ on $\Omega^{\bullet}(\mathrm{St}\,u)$ gives rise to a
differential $d$ on $\mathcal{C}^{p,\bullet}$ for each $p$.

######  Proposition 4.3.

The map $\delta$ is a differential on $\mathcal{C}^{\bullet,q}$ for each
$q\in\mathbb{Z}$. Moreover, the double graded vector space
$\mathcal{C}^{\bullet,\bullet}$ together with $\delta$ and $d$ forms a double
complex in the sense that $d\,\delta=\delta\,d$.

######  Lemma 4.4.

There exist elements $p_{v}\in A^{0}(X),\,v\in V(X)$, such that

$\sum_{v\in V(X)}p_{v}t_{v}^{2}=1.$

###### Proof.

In $A^{0}(X)$ we have the equality

$\sum_{v\in V(X)}t_{v}=1.$

Raise both the sides to a big enough power and obtain the needed equality. ∎

We put $\rho_{v}=p_{v}t_{v}^{2}$.

For an inclusion of simplicial complexes $Y\subset X$ we choose a linear map,
the distinguished “extension”,

$[-]:\Omega^{q}(Y)\to\Omega^{q}(X),$

such that $[\omega]|_{Y}=\omega$. Such an extension exists by Lemma 3.1
applied to $A^{0}(X)$ and Lemma 4.1.

######  Lemma 4.5.

For an inclusion of simplicial complexes $Y\subset X$ and a form
$\omega\in\Omega^{q}(Y)$ we have

$\sum_{v\in V(X)}\rho_{v}[\omega|_{\mathrm{St}\,_{Y}(v)}]|_{Y}=\omega.$

###### Proof.

As $\sum_{v}\rho_{v}=1$ in $A^{0}(X)$ we have

$\sum_{v\in
V(X)}\rho_{v}[\omega|_{\mathrm{St}\,_{Y}(v)}]|_{Y}-\omega=\sum_{v\in
V(X)}\rho_{v}|_{Y}\left([\omega|_{\mathrm{St}\,_{Y}(v)}]|_{Y}-\omega\right).$

We have

$\left([\omega|_{\mathrm{St}\,_{Y}(v)}]|_{Y}-\omega\right)|_{\mathrm{St}\,_{Y}(v)}=\omega|_{\mathrm{St}\,_{Y}(v)}-\omega|_{\mathrm{St}\,_{Y}(v)}=0.$

Hence, by Lemma 4.2 we have
$\rho_{v}|_{Y}\left([\omega|_{\mathrm{St}\,_{Y}(v)}]|_{Y}-\omega\right)=0$. ∎

######  Proposition 4.6.

The complex

${0}$${\mathcal{C}^{-1,q}}$${\mathcal{C}^{0,q}}$${\mathcal{C}^{1,q}}$${\dots}$${\Omega^{q}(X)}$.$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

is exact.

The proof follows the proof of [4, Proposition $8.5$].

###### Proof.

First, notice that

$\mathrm{St}\,v*u=\mathrm{St}\,_{\mathrm{St}\,u}(v).$

For $u\in T_{p}(X)$ and $\omega\in\Omega^{q}(\mathrm{St}\,u)$ by Lemma 4.5 we
have

(1) $\sum_{v\in
V(X)}\rho_{v}[\omega|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}=\omega.$

For $\omega\in\Omega^{q}(Z)$, where $Z$ is a subcomplex of $X$ such that
$\mathrm{St}\,v*u\subset Z$, by Lemma 4.2, we have

(2)
$\rho_{v}\left([\omega]-[\omega|_{\mathrm{St}\,v*u}]\right)|_{\mathrm{St}\,u}=0.$

We construct a cochain homotopy

$K:\mathcal{C}^{p,q}\to\mathcal{C}^{p-1,q}.$

For $p\geq 0$ and $\omega\in\mathcal{C}^{p,q}$ and $w\in T_{p-1}(X)$ put

$(K\omega)_{w}:=\sum_{v\in V(X)}\rho_{v}[\omega_{v*w}]|_{\mathrm{St}\,w}.$

By Lemma 4.2 this map does not depend on the choice of the distinguished
extension.

Let us check that $\delta K+K\delta=1$. For $p\geq-1$ and

$\omega=(\omega_{u})_{u\in T_{p}(X)}\in\mathcal{C}^{p,q}\subset\prod_{u\in
T_{p}(X)}\Omega^{q}(\mathrm{St}\,u),$

where $\omega_{u}\in\Omega^{q}(\mathrm{St}\,u)$, we have

$(\delta
K\omega)_{u}=\sum_{i=0}^{p}(-1)^{i}(K\omega)_{\partial_{i}u}|_{\mathrm{St}\,u}=\sum_{i=0}^{p}(-1)^{i}\sum_{v\in
V(X)}\rho_{v}[\omega_{v*\partial_{i}u}]|_{\mathrm{St}\,u},$

and

$(K\delta\omega)_{u}=\sum_{v\in
V(X)}\rho_{v}[(\delta\omega)_{v*u}]|_{\mathrm{St}\,u}=$ $=\sum_{v\in
V(X)}\rho_{v}[\,\sum_{i=0}^{p+1}(-1)^{i}\omega_{\partial_{i}(v*u)}|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}=$
$=\sum_{v\in
V(X)}\rho_{v}[\omega_{u}|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}+\sum_{v\in
V(X)}\rho_{v}[\,\sum_{i=1}^{p+1}(-1)^{i}\omega_{\partial_{i}(v*u)}|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}\overset{\text{by
(1)}}{=}$ $=\omega_{u}-\sum_{v\in
V(X)}\rho_{v}[\,\sum_{i=0}^{p}(-1)^{i}\omega_{v*\partial_{i}u}|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}$
$=\omega_{u}-\sum_{i=0}^{p}(-1)^{i}\sum_{v\in
V(X)}\rho_{v}[\omega_{v*\partial_{i}u}|_{\mathrm{St}\,v*u}]|_{\mathrm{St}\,u}.$

Hence,

$((\delta K+K\delta)\omega)_{u}=\omega_{u}+\sum_{i=0}^{p}(-1)^{i}\sum_{v\in
V(X)}\rho_{v}\left([\omega_{v*\partial_{i}u}]-[\omega_{v*\partial_{i}u}|_{\mathrm{St}\,v*u}]\right)|_{\mathrm{St}\,u}\overset{\text{by
(2)}}{=}\omega_{u}.$

∎

##  5\. The morphism $P:\Omega^{\bullet}(X)\to A^{\bullet}(X)$

For a simplicial complex $X$, by the universal property of
$\Omega^{\bullet}(X)$, there is a canonical morphism dg-algebras

$P:\Omega^{\bullet}(X)\to A^{\bullet}(X),$

which is the identity in degree $0$.

We denote by $k[0]$ the complex with the $0$th term $k$ and the others zero.
An element of $k$ gives rise to a constant function in $A^{0}(X)$, hence,
there are morphisms of complexes $\epsilon:k[0]\to\Omega^{\bullet}(X)$ and
$\bar{\epsilon}:k[0]\to A^{\bullet}(X)$ such that
$\bar{\epsilon}=P\circ\epsilon$.

######  Proposition 5.1.

For a sequence $u\in T_{p}(X)$, $p\geq 0$, the commutative diagram

${k[0]}$${\Omega^{\bullet}(\mathrm{St}\,u)}$${A^{\bullet}(\mathrm{St}\,u)}$$\scriptstyle{\epsilon}$$\scriptstyle{\bar{\epsilon}}$$\scriptstyle{P}$

consists of quasi-isomorphisms.

###### Proof.

The map $\epsilon$ is a quasi-isomorphism by [2, Corollary $47$]. The map
$\bar{\epsilon}$ is a quasi-isomorphism by [5, Theorem $10.9$]. Hence, the
morphism $P$ is a quasi-isomorphism. ∎

######  Theorem 5.2.

The natural map

$P:\Omega^{\bullet}(X)\to A^{\bullet}(X)$

is a quasi-isomorphism.

###### Proof.

The morphism $P$ on stars gives rise to the maps
$\pi_{p}:\mathcal{C}^{p,\bullet}\to\mathcal{D}^{p,\bullet}$ for each $p\geq
0$. We have the following commutative diagram of non-negative complexes

${0}$${\Omega^{\bullet}(X)}$${\mathcal{C}^{0,\bullet}}$${\mathcal{C}^{1,\bullet}}$${\dots}$${0}$${A^{\bullet}(X)}$${\mathcal{D}^{0,\bullet}}$${\mathcal{D}^{1,\bullet}}$${\dots.}$$\scriptstyle{\delta}$$\scriptstyle{P}$$\scriptstyle{\delta}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\delta}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

The vertical arrows $\pi_{p}$, $p\geq 0$ are quasi-isomorphisms by Proposition
5.1. The first row is exact by Proposition 4.6. The second row is exact by
Proposition 3.3.

Hence, the map $P$ is also a quasi-isomorphism. ∎

######  Remark 5.3.

As was said in the introduction, the paper [7] suggests that Theorem 5.2 is
false. Namely, in [7, Example $3.8$], one considers the simplicial complex $X$
corresponding to the boundary of a triangle on the vertices $1$, $2$, $3$. The
dg-algebra $\Omega^{\bullet}(X)$ is generated by the elements $t_{1}$,
$t_{2}$, $t_{3}$ modulo the dg-ideal generated by $t_{1}+t_{2}+t_{3}-1$ and
$t_{1}t_{2}t_{3}$. Next, the author considers the form
$t_{1}^{2}t_{2}^{2}dt_{3}$ and claims that this form is not zero. However,
this form is zero, which can be seen as follows: applying the differential $d$
to the equality $t_{1}t_{2}t_{3}=0$ we get

$t_{2}t_{3}dt_{1}+t_{1}t_{3}dt_{2}+t_{1}t_{2}dt_{3}=0.$

Next, we multiply this equality by $t_{1}t_{2}$ and get

$0=t_{1}t_{2}^{2}t_{3}dt_{1}+t_{1}^{2}t_{2}t_{3}dt_{2}+t_{1}^{2}t_{2}^{2}dt_{3}=t_{1}^{2}t_{2}^{2}dt_{3}.$

## References

  * [1] D. Arapura and S.-J. Kang, Kähler–de Rham cohomology and Chern classes, Communications in Algebra, 39 (2011), pp. 1153–1167.
  * [2] I. Baskov, The de Rham cohomology of soft function algebras.  https://arxiv.org/abs/2208.11431, 2022.
  * [3] L. J. Billera, The algebra of continuous piecewise polynomials, Advances in Mathematics, 76 (1989), pp. 170–183.
  * [4] R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer, 1982.
  * [5] Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Springer, 2012.
  * [6] Y. Félix, B. Jessup, and P.-E. Parent, The combinatorial model for the Sullivan functor on simplicial sets, Journal of Pure and Applied Algebra, 213 (2009), pp. 231–240.
  * [7] F. Gómez, Simplicial types and polynomial algebras, Archivum Mathematicum, 38 (2002), pp. 27–36.
  * [8] P. Griffiths and J. Morgan, Rational homotopy theory and differential forms, Birkhäuser, 1981.
  * [9] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publications Mathématiques de l’IHÉS, 29 (1966), pp. 95–103.
  * [10] D. M. Kan and E. Y. Miller, Sullivan’s de Rham complex is definable in terms of its 0-forms, Proceedings of the American Mathematical Society, 57 (1976), pp. 337–339.
  * [11] E. Kunz, Kähler differentials, Friedr. Vieweg & Sohn, 1986.
  * [12] D. Sullivan, Differential forms and the topology of manifolds, Manifolds Tokyo, (1973), pp. 37–49.
  * [13] , Infinitesimal computations in topology, Publications Mathématiques de l’IHÉS, 47 (1977), pp. 269–331.

St. Petersburg Department of Steklov Mathematical Institute
of Russian Academy of Sciences

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