# Communication network model of the immune system identifies the impact of
interactions with SARS-CoV-2 proteins

Swarnavo Sarkar<EMAIL_ADDRESS><EMAIL_ADDRESS>National
Institute of Standards and Technology, Gaithersburg, MD 20899, USA

###### Abstract

Interactions between SARS-CoV-2 and human proteins (SARS-CoV-2 PPIs) cause
information transfer through biochemical pathways that contribute to the
immunopathology of COVID-19. Here, we present a communication network model of
the immune system to compute the information transferred by the viral proteins
using the available SARS-CoV-2 PPIs data. The amount of transferred
information depends on the reference state of the immune system, or the state
without SARS-CoV-2 PPIs, and can quantify how many variables of the immune
system are controlled by the viral proteins. The information received by the
immune system proteins from the viral proteins is useful to identify the
biological processes (BPs) susceptible to dysregulation, and also to estimate
the duration of viral PPIs necessary for the dysregulation to occur. We found
that computing the drop in information from viral PPIs due to drugs provides a
direct measure for the efficacy of therapies.

## Introduction

COVID-19, the disease caused by severe acute respiratory syndrome coronavirus
2 (SARS-CoV-2), has multiple routes for harming human biology. One mechanism
is through protein-protein interactions (PPIs) between SARS-CoV-2 and human
proteins (SARS-CoV-2 PPIs) [1]. The viral PPIs can alter biochemical functions
of human proteins, the effect of which can propagate through the cascade of
biochemical reactions informing the abundance of other proteins that do not
directly interact with the viral proteins. The information transfer caused by
SARS-CoV-2 PPIs may elucidate some of the more complex consequences of
COVID-19, like long lasting immunological dysfunction [2, 3, 4], dysregulation
of myeloid response [5], lymphopenia [6, 7], and more. For a comprehensive
assessment of COVID-19, we need to quantify the information about SARS-CoV-2
PPIs that is communicated to the immune system proteins and the sensitivity of
biological processes (BPs) to the amount of transferred information. The
fundamental framework for communication is information theory [8], which has
been used to quantify the efficacy of biochemical signaling [9, 10, 11].
Information theory uses entropy to measure the dependence between stochastic
variables, and all biochemical entities are stochastic at the cellular level.
By bringing the fundamental information-theoretic model of biochemical
signaling to the immune system, we can computationally determine the
ramifications of COVID-19 and other infections. Computing the information
transfer in the immune system due to viral PPIs also provides a framework for
in silico evaluation of therapies [12].

Here, we developed a communication network model of the immune system using
pathways and interaction data to compute the information transferred by viral
PPIs, and to subsequently identify potentially dysregulated BPs and efficacy
of immunotherapies. We modeled biochemical reactions and human protein-protein
interactions as channels transferring information [8, 13] between the physical
entities (proteins, complexes, metabolites, etc.) that are part of the immune
system. In this case study, the SARS-CoV-2 protein interaction map [1]
provides the sources of information in the immune system communication
network. To determine the information transferred to the immune system
proteins, we use an algorithm that is sensitive to the inter-connectivity of
the immune system network [14], resulting in the interference of information
reaching a physical entity from multiple pathways. We found that the
information transfer from SARS-CoV-2 PPIs depends on the reference state of
the immune network (i.e., the state in the absence of SARS-CoV-2 PPIs), which
subsequently determines the BPs that can be dysregulated due to communication.
Using the large deviations theory [8, 15], we estimate when a BP becomes
significant for dysregulation due to communication from SARS-CoV-2 PPIs.
Furthermore, we directly compute the reduction in information from viral PPIs
due to candidate therapies, providing a new measure to rank the efficacy of
drugs.

## Immune system communication network

To construct the communication network model of the immune system, we modeled
the immune system as a graph $G=(V,E)$ made of nodes, $V$, and edges, $E$. We
obtained the nodes and the edges from two databases: the Reactome database for
the physical entities participating in the immune system and the reactions
interconnecting them [16], and the Human Reference Interactome database for
the pairwise interactions between proteins [17]. The physical entities form
the nodes of the network, whereas the reactions and the interactions
contribute as edges (LABEL:sec:_network_construction). The immune system
communication network consists of $|V|=3011$ nodes, with 1141 protein nodes
(Fig. 1a, Table LABEL:table:_node_composition). We represented the state of
each node as a Bernoulli random variable: if $v_{i}\in V$ is a network node,
then the probabilistic state of the node is $\\{P(v_{i}=1),P(v_{i}=0)\\}$,
where the abundance (no. of molecules) of $v_{i}$ is coarse-grained into a
binary variable, 1 (abundant) and 0 (deficient). The state of the entire
immune system network is a $2|V|$-dimensional vector $X$, where the $i^{th}$
2-dimensional block of $X$ contains the state of $v_{i}$. The reactions and
interactions connect pairs of nodes with input-output relationships
(LABEL:sec:_network_construction).

To simulate the flow of information in the immune system, we constructed a
global transition matrix to iteratively update the network state due to
communication. We modeled each edge as a binary information channel, with a
2-by-2 transition matrix $P(output|input)$, which communicates the state of
the input to the output (LABEL:sec:_channel_model). Then we used the
connectivity of the network $G$, and unbiased interference of information, to
assemble the transition matrix of individual edges into the transition matrix
for the entire network $C_{G}$ (LABEL:sec:_channel_model) [14]. Information
transfer through the network updates the state of the network as

$X^{(k+1)}=C_{G}X^{(k)}$ (1)

where $X^{(k)}$ is the network state after $k$ steps of information
propagation. If the initial state is $X^{(0)}$, then after $k$ steps of
information transfer the state is $X^{(k)}=C_{G}^{k}X^{(0)}$. The stationary
state of the network satisfies $X=C_{G}X$. There are 89 source nodes in the
network (i.e., nodes that only send but do not receive information), the state
of these nodes determine the reference state of the immune system
communication network in the absence of other information sources, like SARS-
CoV-2 PPIs. Current limitations of the communication network model include,
incompleteness and errors in the pathways and interactions data, a binary
channel to model the reactions and interactions, and unbiased interference of
information. Proteomic, metabolomic, and rate constants data for human cells
can provide data-driven parameters for channel bit size and interference bias.
We expect to progressively integrate such data in the future versions of the
model.

##  Information transferred by SARS-CoV-2 PPIs

To evaluate the information transferred by SARS-CoV-2 PPIs, we assumed that
interactions with SARS-CoV-2 proteins diminish the function of human proteins.
So, the impact of SARS-CoV-2 PPIs is a reduction in the functional or the
effective abundance of the associated human proteins. For each of the proteins
that directly interact with SARS-CoV-2 proteins (SARS-CoV-2 PPI node), we
constrained the state of that protein node to $\\{0,1\\}$, or effectively to
low abundance. We need experimental studies to assess the exact decline, if at
all, in a protein’s function due to SARS-CoV-2 PPIs (as available for
interactions with the ORF9b protein [18]). However, we can computationally
test all possible impact on the protein function, i.e., maximum decline in
function (constraint $\\{0,1\\}$), no effect on the function (no constraint),
or maximum increase in function (constraint $\\{1,0\\}$), or any intermediate
constraint between $\\{1,0\\}$ and $\\{0,1\\}$. We used the constraint
$\\{0,1\\}$ to compute the consequence of maximum decline in a protein’s
function due to SARS-CoV-2 PPIs.

To quantify the information transferred by SARS-CoV-2 PPIs, we first computed
the reference stationary state $X_{\mathrm{ref}}$ in the absence of the SARS-
CoV-2 proteins, and then constrained the SARS-CoV-2 PPI nodes as described
above to compute a new stationary state $X$ using Eq. (1)
(LABEL:sec:_PPI_set). The information transferred to a single network node
$v_{i}$ due to the SARS-CoV-2 PPIs is given by the relative entropy (or the
Kullback-Leibler divergence) in bits as [8]

$H_{\mathrm{ref}}(v_{i})=\sum_{k\in\\{2i,2i+1\\}}X[k]\log_{2}\frac{X[k]}{X_{\mathrm{ref}}[k]}\quad.$
(2)

The network relative entropy $H_{\mathrm{ref}}(X)$ is the sum of the relative
entropy of all the unconstrained network nodes. We computed the information
transferred due to SARS-CoV-2 PPIs for multiple reference states (Fig. 1b). In
Protocol 1 all the source nodes are at the maximum entropy state
$\\{0.5,0.5\\}$, which results in a $X_{\mathrm{ref}}$ where all the other
nodes are also at maximum entropy. Then for each SARS-CoV-2 protein we
constrained the corresponding subset of SARS-CoV-2 PPI nodes at $\\{0,1\\}$ to
compute the stationary state $X$ using Eq. (1) (PPI set,
LABEL:table:_PPI_set). In this protocol, we found that the SARS-CoV-2 proteins
transfer between 0.15 bits (ORF3a protein) to 16.2 bits (M, membrane protein)
of information (Fig. 1b). Each bit of information transfer is effectively a
reduction in the number of variables in the network. For example, during the
interaction with SARS-CoV-2 NSP2, the network has a total of 2921
unconstrained nodes, therefore the maximum possible $H_{\mathrm{ref}}(X)$ is
2921 bits, at which the NSP2 protein completely determines the state of the
entire immune network. Whereas, the actual $H_{\mathrm{ref}}(X)$ for NSP2 is
0.97 bits, or effectively it controls 1 out of the 2921 variables. A higher
network relative entropy means that the particular SARS-CoV-2 protein is a
more influential regulator of the immune system. When all source nodes are at
high abundance $\\{1,0\\}$, then the SARS-CoV-2 PPIs produce a higher relative
entropy (Protocol 2 in Fig. 1b). When all source nodes are at $\\{0,1\\}$ then
the information transferred is lower (Protocol 3 in Fig. 1b). Or, when the
effect of SARS-CoV-2 PPIs on protein function is opposite to abundances in the
reference state, the information transferred is higher. The variability in
$H_{\mathrm{ref}}(X)$ for the three protocols show that SARS-CoV-2 PPIs
signaling is dependent on the proteomic and metabolomic profile (or
abundances) of infected individuals. However, in a realistic scenario all the
source nodes are unlikely to be in the same state. So, we computed the
distribution of the network relative entropy using a Monte Carlo (MC)
simulation. For each MC sample, we constrained each source node to a state
uniformly distributed between $\\{1,0\\}$ and $\\{0,1\\}$ to first compute the
reference state and then the state under SARS-CoV-2 PPIs. The median network
relative entropy is equal to the value computed in Protocol 1, because
unbiased interference of information drives the nodes closer to the maximum
entropy state. The dispersion in information transfer, or the inter-quartile
range of $H_{\mathrm{ref}}(X)$, is 4 bits for the M protein, but is around 1
bit or less for all the other SARS-CoV-2 proteins. Protocol 1 is particularly
useful because it shows the information transferred only by SARS-CoV-2 PPIs,
without assistance from the other source nodes in the network, and quantifies
how many variables of the immune system are controlled by the viral proteins.

## Effect on biological processes

To determine the effect of communication from SARS-CoV-2 PPIs on biological
processes (BPs), we first identified the immune system proteins that receive
$H_{\text{ref}}(v_{i})\geq 0.01$ bits of information (or the _communicated
set_ of proteins) (LABEL:sec:_communicated_proteins). For each SARS-CoV-2
protein, we added the communicated set with the PPI set, and performed Gene
Ontology overrepresentation analysis for BPs [19, 20, 21] (Fig. 2,
LABEL:sec:_ORA). Including the communicated set not only identifies more BPs
as significant (Fig. 2), but also shows that more than one SARS-CoV-2 protein
can influence the same BP due to information transfer (e.g. immune effector
process and cell activation in Fig. 2). The effect of COVID-19 on cytokine
production and inflammation is well known [22, 23]. But several of the other
BPs that are significant due to communication have also been reported. These
include, rapid decay in humoral immunity [24, 25], increased apoptosis in
infected cells [26, 27], localization of organelles, especially endoplasmic
reticulum, endosome, and lysosome [28, 29], modulation of catabolic recycling
processes [30], and protein import into the mitochondria [31], to name a few.
Additionally, dysregulation of some of the cell activation processes due to
COVID-19, release of myeloid dendritic cells [5, 32] and lymphocyte production
[6, 7], have also been reported.

Using the information transferred by SARS-CoV-2 PPIs we can address the
question: how might the significant BPs depend on the duration of SARS-CoV-2
PPIs? We answer this question using Protocol 1, where the reference state of a
protein is $\\{0.5,0.5\\}$. If a protein is actually in the reference state,
then after $N$ responses (or samples) of the protein abundance the empirical
state can be away from $\\{0.5,0.5\\}$ due to stochasticity. From large
deviations theory, the probability of observing an empirical state with a
relative entropy greater than or equal to $H_{\mathrm{ref}}(v_{i})$ after $N$
responses is $2^{-H_{\mathrm{ref}}(v_{i})N}$ [15, 8] (LABEL:sec:_en-ORA). To
account for this stochastic deviation from the reference state, which can
occur in the absence of SARS-CoV-2 PPIs, we computed the $Q$-value of BPs as a
function of the number of responses. We performed a Monte Carlo simulation
where for each MC sample we eliminated proteins from the communicated set with
the probability $2^{-H_{\mathrm{ref}}(v_{i})N}$ and computed a set of raw
$p$-values for the GOBP terms. Then we performed the multiple hypothesis
correction on the averaged set of $p$-values to identify the significant BPs
as a function of $N$ (LABEL:sec:_en-ORA). At shorter duration ($N=10$) several
BPs may not appear susceptible to dysregulation, but at longer duration
($N=1000$) the full consequence of communication from SARS-CoV-2 PPIs is
revealed (Fig. 3a). Different SARS-CoV-2 proteins can take different number of
responses to effect the same BP. So, computing the $Q$-$N$ curves for each of
the significant BPs is more insightful compared to only determining the
significant BPs after a long duration of SARS-CoV-2 PPIs (Fig. 3b for NSP12
and ORF9c, figures for the other SARS-CoV-2 proteins in LABEL:sec:_en-ORA). We
found that the effect of NSP12 on necrotic cell death can occur earlier than
on cell activation, and the effect of ORF9c on defense response can occur
earlier than on protein import. The $Q$-$N$ curves (Fig. 3b) may answer
clinically relevant questions, like the consequences of any delay in
administering protease inhibitors during COVID-19. Protease inhibitors, like
nirmatrelvir/ritonavir (sold as Paxlovid), reduce viral replication and
consequently decrease the duration of SARS-CoV-2 PPIs. Therefore, using the
$Q$-$N$ curves we can identify the BPs that can get dysregulated due to any
delay in the use of protease inhibitors.

## Evaluation of candidate therapies

To use the communication network model for evaluating candidate therapies, we
computed the reduction in the information transfer from SARS-CoV-2 PPIs due to
drugs. We demonstrated this technique using the maximum entropy reference
state (as in Protocol 1 in Fig. 1b) and then applying the constraint
$\\{0,1\\}$ on all SARS-CoV-2 PPI nodes together (all proteins in Table
LABEL:table:_PPI_set), which produces a total network relative entropy of
$65.4$ bits. The Reactome database contains 19 drugs that interact with the
immune system, majority of which are immunomodulators [33]. We constrained
each of the drugs to the state $\\{1,0\\}$, one at a time, and computed the
subsequent change in network relative entropy, $\Delta H_{\mathrm{ref}}(X)$
(Fig. 4a). The maximum net reduction in the SARS-CoV-2 PPIs signaling was 2
bits (for ibrutinib and acalabrutinib) and the maximum net increase was 3 bits
(for omalizumab). Or, at best some of the drugs can effectively free 2 out of
the 65 variables of the immune system that are under the control of SARS-CoV-2
PPIs. 12 out of the 19 drugs produced less than $\pm 1$ bit change in the
network relative entropy, indicating that majority of these 19 drugs have
minimal effect in countering the signaling from SARS-CoV-2 PPIs. However, a
drug can reduce the information transfer from SARS-CoV-2 PPIs in some nodes of
the immune system (drop), but it can also cause additional information
transfer in other nodes of the immune system (gain). So, for a more nuanced
evaluation of the efficacy of a drug, we identified the network nodes that
have a drop in relative entropy due to a drug and in which nodes there is a
gain, to obtain the cumulative drop/gain values (Fig. 4b). Sorting by the drop
component instead of $\Delta H_{\mathrm{ref}}(X)$ produces a different ranking
of the drugs. We cannot neglect the gain component for new drugs, as we need
to scrutinize the proteins that are receiving the gained information and the
subsequent influence on BPs. However, if a drug is known to be safe, then we
can neglect the gain component and use the drug with the largest drop value
(Fig. 4b). Among the drugs that have large gain values, Janus Kinase (JAK)
inhibitors, which inhibit cytokine activity, were considered to be safe [34,
35], but their safety profile is being re-evaluated [36]. Omalizumab, an anti-
Immunoglobulin E drug, has several side effects, including anaphylaxis [37].
JAK inhibitors have the best drop values, and they have been shown to increase
survival during severe COVID-19 in clinical trials, but insignificant for
reducing hospitalization [38, 39, 40]. Bruton’s tyrosine kinase (BTK)
inhibitors, which impede the B-cell receptor pathway, have the second best
drop values, and ibrutinib has been claimed to reduce pulmonary injury due to
COVID-19 in a clinical trial [41]. Interleukin 6 (IL-6) inhibitors emerge as
comparatively less effective in our analysis, which is in accordance with the
results of clinical trials – IL-6 inhibitors are significantly beneficial only
when administered with glucocorticoids in clinical trials [42, 43]. But the
limitations in our model is probably underestimating the efficacy of
tocilizumab, which is one of the recommended treatments for critical COVID-19
[44]. As for the other drugs, the relatively low efficacy of Interleukin-1
(IL-1) inhibitors agrees with the clinical data [45, 46]. Programmed cell
death protein 1 (PDCD1) inhibitors, which are immune checkpoint inhibitors,
are hypothesized to treat COVID-19 induced lymphopenia [47], but we did not
find any study to support the claim. Omalizumab is still in a clinical trial
for COVID-19 treatment [48]. The current NIH treatment guideline for COVID-19
recommends only baricitinib and tofacitinib among the JAK inhibitors and
mostly against the BTK, IL-6, and IL-1 inhibitors, except for the cautious use
of tocilizumab and no recommendation for or against anakinra [33].

We similarly computed the effectiveness of artificially controlling the
expression level of proteins to reduce the information transfer from SARS-
CoV-2 PPIs. One at a time, we set the immune system proteins (except the SARS-
CoV-2 PPI nodes) at high abundance $\\{1,0\\}$ and computed the subsequent
change in the network relative entropy. We used artificial overexpression of
proteins in this study because it is opposite to the effect of SARS-CoV-2 PPIs
on protein function in our model. We found that 638 out of the 1122 proteins
that we surveyed produced a better $\Delta H_{\mathrm{ref}}(X)$ than any of
the 19 drugs (_i.e_ , more than 2 bits, Fig. 4c). In particular, we found 23
proteins that can reduce the signaling from SARS-CoV-2 PPIs by 28 to 30 bits,
and this set contains 3 IFITM (interferon-induced transmembrane) proteins.
Interestingly, artificial overexpression of IFITM proteins has been shown to
block SARS-CoV-2 infection in vitro, even though endogenous overexpression of
IFITM proteins supports the infection [49]. PSMB8 (Proteasome subunit beta
type-8 protein, involved in antigen processing) has been reported to cause
milder infection under endogenous overexpression using data from lung samples
[50]. Several of the other 23 proteins identified in this analysis (_e.g._
MUC1, MX1, MX2, IFIT1, XAF1, EGR1) have been reported to be overexpressed
during COVID-19, depending on the severity of the disease [51, 52, 53, 54,
55]. We separated the drop/gain contributions to $\Delta H_{\mathrm{ref}}(X)$
for each of the proteins (Fig. 4d). The 23 proteins that produced the best
$\Delta H_{\mathrm{ref}}(X)$ remain close to the top hundred even after
ranking the proteins by the drop component. So, it is possible to avoid the
proteins with high gain values without compromising on the reduction to SARS-
CoV-2 PPIs signaling. Since controlled expression of the proteins listed in
Fig. 4c can reduce the information transfer considerably better than any of
the drugs, we may be able to create more effective treatments for SARS-CoV-2
PPIs compared to the ones in Fig. 4b.

## Conclusion

In summary, we developed a communication network model of the immune system to
compute the information received by the various components of the system
(proteins, complexes, etc.) through multiple biochemical pathways. We
demonstrated this model by computing the information transferred by SARS-CoV-2
proteins to the immune system due to their interaction with the human
proteins, which reveals the ranking of the viral proteins based on their
ability to control the immune system. Since the information transferred by the
viral PPIs is sensitive to the reference state of the immune system, we can
estimate the variability in the immunopathology of COVID-19 due to the
proteomic and metabolomic profile of infected individuals. Enrichment analysis
using the set of proteins that receive communication from SARS-CoV-2 PPIs
leads to the potentially affected biological processes, many of which have
been reported in the experimental and clinical data on COVID-19. Moreover, we
can estimate the rate at which various biological processes approach
dysregulation, or the $Q$-$N$ curves, as a function of the duration of SARS-
CoV-2 PPIs. Perhaps the most interesting feature of the communication network
model is the ability to compute the efficacy of therapies in reducing the
information from viral PPIs and also the unwanted information transfer caused
by the treatment, providing a new computational framework for drug selection.
The top ranked drugs and proteins from our computations find support in the
clinical and in vitro data for COVID-19, in spite of the simplifying
assumptions in the model.

The computational software that we have developed can construct similar
communication network models for other modules in the Reactome database,
_e.g._ , Metabolism, Programmed Cell Death, Hemostasis, and can also combine
multiple modules. The information transfer calculation is not restricted to
viral PPIs only. We can use differential omics data to conceptualize reference
and perturbed states, and can compute the information transfer due to various
types of diseases and treatments.

## Acknowledgement

We would like to thank Mehdi Bouhaddou, Farhad Maleki, and the Reactome
helpdesk team, especially Robin Haw, for helpful information during this work.
We would like to thank Charles H. Camp, William Valiant, and Nancy J. Lin for
their comments on the manuscript.

## Code Availability

The source code to construct the communication network from the Reactome and
the Interactome databases and to perform the subsequent computations is
available at github.com/sarkar-s/core.

(a)

(b)

Figure 1: Information transfer in the immune system due to SARS-CoV2 PPIs. (a)
Communication network model of the immune system. Green nodes are proteins;
blue nodes are complexes; orange nodes are molecules like ATP, ADP, or ions
(Table LABEL:table:_node_composition). The red nodes are proteins that
directly interact with the SARS-CoV-2 proteins, these nodes act as information
sources in the communication network model of the immune system. The black
edges are reaction like events from the Reactome database [16] and the green
edges are protein-protein interactions from the Human Reference Interactome
database [17]. (b) Network relative entropy induced by each SARS-CoV-2
protein. NSP: non structural proteins; ORF: open-reading frames for accessory
factors; E: envelope protein; M: membrane protein; and N: nucleocapsid
protein. In Protocol 1 all source nodes are set at the maximum entropy state,
in Protocol 2 they are at maximum abundance $\\{1,0\\}$, and in Protocol 3
they are at minimum abundance $\\{0,1\\}$. The bottom-most subplot in (b)
shows the distribution in the network relative entropy when each source node
is uniformly distributed between maximum and minimum abundance. The
distribution in network relative entropy was computed from $10^{4}$ samples
(we compared the result against $4\times 10^{4}$ samples for convergence). The
horizontal orange line in each box is the median, the box bounds the
interquartile range (IQR), and the caps bound 1.5IQR below and above the first
and the third quartiles, respectively. The red dots are the network relative
entropy values from Protocol 1, which coincide with the median values. Figure
2: Significant biological processes due to communication from SAR-CoV-2 PPIs.
Each grid plot is the result of a Gene Ontology (GO) overrepresentation
analysis for a set of proteins associated with each of the SARS-CoV-2
proteins. The enrichment analysis was done using the GOBP gene sets in the
MSigDB database (LABEL:sec:_ORA) [56, 57]. The significant BPs are labeled
using their adjusted $p$-value ($Q$) after multiple hypothesis testing. The
first block of $Q$ values (shaded gray) is for the PPI set for each SARS-CoV-2
protein, and the remaining three blocks are for the sum of the PPI and
communicated sets. We identified the significant BPs using the hypergeometric
test and a false-discovery rate of 1% [58], which is the same value used by
the authors of the SARS-CoV-2 protein interaction map [1]. The top 3
significant BPs for each SARS-CoV-2 protein in Protocol 2 were selected for
this figure (top BPs), because Protocol 2 produces the highest amount of
information transfer in the network. We show the $Q$ values only for the top
BPs using the proteins from the PPI set, Protocol 1, and Protocol 3, to
compare the effect of information transfer. The complete list of all
significant BPs for each protocol is in LABEL:sec:_ORA.

(a)

(b)

Figure 3: Significant biological processes as a function of the duration of
SARS-CoV-2 PPIs. (a) Each block of $Q$ values is from enrichment analysis
after a finite number of responses. $N$ is directly proportional to the
duration of SARS-CoV-2 PPIs, or the duration of SARS-CoV-2 PPIs $\approx
N\times$ average response time of immune system reactions. The set of BPs
chosen for this plot are the same as in Fig. 2. The $Q$ values were computed
from a Monte Carlo simulation with $10^{4}$ samples (LABEL:sec:_en-ORA). For
each MC sample the communicated set of proteins is a random variable, which
produces a set of $p$-values for the GOBP terms. The $N$-dependent $Q$ values
were computed by applying the Benjamini-Hochberg procedure to the average
$p$-values from the MC simulation. (b) Finite response dependent $Q$ values
for the significant BPs associated with SARS-CoV-2 NSP12 and ORF9c proteins.
Similar plots for other SARS-CoV-2 proteins are in LABEL:sec:_en-ORA. The
horizontal gray line shows the false-discovery rate used to identify the
significant BPs, which is the same value used by Gordon et. al. [1].
Intersection of the gray line and the $Q$-$N$ curve for a BP estimates the
number of responses after which the BP becomes significant for dysregulation.

(a)

(b)

(c)

(d)

Figure 4: Countering the information transfer from the SARS-CoV2 PPIs using
therapies. (a) Effectiveness of chemical and protein drugs in reducing the
information transfer from SARS-CoV-2 PPIs. The black vertical line is the
network relative entropy when all the protein nodes in SARS-CoV-2 PPIs
(LABEL:table:_PPI_set) are at $\\{0,1\\}$. The solid dots are network relative
entropies when a drug is present in high abundance. The colorbar is scaled
according to the net change in network relative entropy due to a drug. (b) The
drop (blue) and gain (red) components of $\Delta H_{\mathrm{ref}}(X)$ due to
drugs. $\Delta H_{\mathrm{ref}}(X)$ in (a) is the sum of the red and the blue
bars. The list of drugs is sorted by the drop value instead of $\Delta
H_{\mathrm{ref}}(X)$. Data for (a) and (b) are in SIdata5. (c) Effectiveness
of artificial overexpression of immune system proteins in reducing information
transfer from the SARS-CoV-2 PPIs. The horizontal line is the network relative
entropy when all the protein nodes in SARS-CoV-2 PPIs (LABEL:table:_PPI_set)
are at $\\{0,1\\}$. The immune system proteins, 1122 in total, are sorted in
the ascending order of $\Delta H_{\mathrm{ref}}(X)$. The list of sorted
proteins is in SIdata6. The green rectangle highlights the 23 proteins that
reduces the information transfer from SARS-CoV-2 PPIs by 28 to 30 bits. The
names of these proteins are listed in the text box. (d) The drop (blue) and
gain (red) in network relative entropy due to artificial overexpression of
proteins. The 1122 immune system proteins are sorted by their drop value. The
green box highlights the region where the 23 most effective proteins from (c)
are located. Protein ranking by the $\Delta H_{\mathrm{ref}}(X)$ drop
component is in SIdata6.

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