# Network medicine in ovarian cancer: Topological properties to drug discovery

Keilash Chirom1,2, Md. Zubbair Malik1∗, Pallavi Somvanshi1∗ and R.K. Brojen
Singh1 R.K.B.S<EMAIL_ADDRESS>(Corresponding author),
M.Z.M<EMAIL_ADDRESS>(Co-corresponding author),
P.S<EMAIL_ADDRESS>(Co-corresponding author) 1School of Computational
$\&$ Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067,
India.
2Department of Biotechnology, TERI University, New Delhi-110070, India.

###### Abstract

The investigation of topological properties of ovarian cancer network (OCN)
and the roles of hubs involved in it by digging the network at various levels
of organization are important to understand how OCN is organized to understand
disease states. The OCN constructed from the experimentally verified genes
exhibits fractal nature in the topological properties of the network and
deeply rooted communities. Also, the network properties at all levels of
organization obey one parameter scaling law which lacks centrality lethality
rule. We then showed that $\langle k\rangle$ can be taken as a scaling
parameter, where, power law exponent can be estimated from the ratio of
network diameters, $\lambda=1+\rho\left[\frac{d_{c}}{d_{k}}\right]$. The
betweenness centrality shows two distinct behaviors one shown by high degree
hubs and the other by segregated low degree nodes. The $C_{B}$ distribution
follows power law behavior with the exponent connected to exponents of
distributions of high and low degree nodes by,
$\theta\sim\left[1+\frac{1}{2}\left(\frac{1}{\epsilon_{s}}+\frac{1}{\epsilon_{h}}\right)(\lambda-1)\right]$.
Absence of rich-club formation leads to the missing of a number of attractors
in the network causing formation of weakly tied diverse functional modules to
keep optimal network efficiency. The hubs knockout experiment shows that
provincial hubs take major responsibility to keep network integrity and
organization. The identified key regulators are found to be provincial and
connector hubs. Further, two key regulators, EPCAM and CD44 are found to be
maximally over expressed at various cancer stages (II-IV). They are also
positively correlated with immune infiltrates (CD4+ T cells). Finally, few
potential drugs are identified related to the key regulators.

Keywords: Ovarian cancer; Fractal, Hierarchical network; Topological
properties; Provincial hubs.

Introduction
Cancer is generally believed to be genetic disease Kolch  and interferes a
number of interacting spatio-temporal signaling gene regulatory networks
(SGRN), which are responsible for molecular basis communications, cellular and
other processes in the whole complex cellular networks Erwin . Since the SGRN
are composed of basic motif circuits Milo , the overall signal processing by
them is the integration of the signals by these circuits Shoval  which may
reflect fractal or scale-free properties in the network organization with
hierarchical features Barabasi  given by self-affine process Calvert  of any
property $\mathcal{F}(x)$ of a scale factor $\alpha$:
$\frac{\mathcal{F}(\alpha x)}{\mathcal{F}(x)}=\Gamma(\alpha)\sim\alpha^{D}$,
such that,
$\Gamma(\alpha_{1}\alpha_{2}...\alpha_{N})=\Gamma_{1}(\alpha_{1})\Gamma_{2}(\alpha_{2})...\Gamma_{N}(\alpha_{N}):\Gamma_{k}(\alpha_{k})\sim\alpha_{k}^{D_{k}}$
with $0\leq\alpha,\alpha_{1},\alpha_{2},...,\alpha_{N}\leq 1$, where,
$D,D_{k}$ are self-similarity dimension. Hence, the static and dynamical
changes in the SGRN due to changes in signal processing when the cellular
system move from normal to disease state are reflected in the topological
properties of the network Boccaletti . Because the changes in the signal
patterns in the network is driven by genetic and epigenetic perturbations
which alter significant rewiring of the these signaling networks Califano ;
Schramm . This perturbation may enhance or diminish the functional activity of
a gene or group of genes causing significant changes in the signaling patterns
leading to variations in the cellular phenotype Creixell  which may exhibit in
the topological properties of the network Teschendorff . On the other hand,
Kauffman’s cancer attractor theory Kauffman ; Huang  developed on the basis of
his works and earlier works of Max Delbruck Delbruck , and Jacob and Monod
Monod , is another important theory to explain the disruption in signaling
mechanism at cancer state Huang . The theory claims that the interacting genes
in each SGRN spatio-temporally regulate among themselves to reach an
equilibrium state or attractor which corresponds to balancing of their gene
expression patterns Kauffman . Hence, the state of a complex network having
$N$ interacting genes with their expression levels $x_{1},x_{2},...,x_{N}$ at
any instant of time ${}^{\prime}t^{\prime}$ can be defined as an
$N$-dimensional vector $\vec{\Lambda}(t)=[x_{1},x_{2},...,x_{N}]^{-1}$ in the
state space Huang . In this state space, there could be a large number of
attractors, where, each attractor corresponds to an equilibrium state $S^{*}$
surrounded by non-stationary unstable states Kauffman ; Kauffman1  and encodes
specific genetic activities Huang . The attractors could be of two types, one
may correspond to stable network corresponding to normal mature cell type, and
the other may correspond to cancer attractor separated by epigenetic barrier
which can be lowered by mutation load. Hence, in the state corresponding to
cancer attractor, the property of the signal processing mechanism in the
network will be quite different and may be measured in the topological
properties of the network. It has been proposed that signaling entropy Gomez ,
which can be calculated from local signaling entropy of the network
Teschendorff , could be one potential parameter to characterize the variations
in the signaling patterns in the network. However, capturing these altered
signaling patterns with network topological properties and their
quantification are still debatable and not fully understood yet.

Ovarian cancer is known as silent killer because early diagnosis of this
cancer type is quite difficult and generally more than 70% cases are not able
to be detected until at significantly advanced stage Jayson ; Carlson . One
reason could be inability to recognize the symptoms of this cancer because the
data indicates that the origin of this cancer is generally from fallopian tube
and not from ovaries Mallen . Hence, this cancer type is one of the most
inheritable cancers, the fifth cause of cancer in women and the fourth cause
of women death in developed countries Berns ; Flaum . Further, since genetic
instability is the hallmark of cancer disease that causes genetic
heterogeneity triggering neoplastic progression Hanahan , it becomes quite
difficult to which key gene/genes would be targeted for possible cure of
cancer diseases Merlo . However, identification of key genes of ovarian cancer
should be done systematically from various sensitive experimental and
theoretical perspectives because those identified key genes are the hopes of
possible drug targets in systematic drug design that may lead to cure and
prevent from this dreaded disease. There have been few techniques to identify
the key regulators, namely, network theory which was used to identify the five
novel key genes (AKT1, KRAS, EPCAM, CD44 and MCAM) in ovarian cancer network
Malik , new biomarker (CREB1 gene) that was proposed by analyzing various
databases with experiments Chia , and GWAS (genome-wide association studies)
based identification of biomarkers specially for African women Manichaikul
etc.

In a particular network, hubs (high degree nodes), specially key regulators,
are the main agents which are involved in many signaling processes, in
maintaining topological properties, stability and self-organization of the
network Barabasi ; Albert . For instance, it has been found that a group of
important hub genes involved in prostate cancer are significantly reduced
their gene expressions and hence quite disturb in their regulatory mechanisms
and signal processing Ribarska , regulatory mechanism of a group of genes in
few pathways (eg glucose metabolism) in Lung cancer are found to be disturbed
Vanhove , significant dysregulation of signaling hubs of lipid-regulated genes
was found in Hepatocellular carcinoma Lee  etc. If the cancer cause
dysregulation or disturbance in the signaling hubs in the complex gene
regulatory network, then there should be significant changes in the
topological properties of the network. The question is how to capture the
changes in the information patterns in the genes and associated networks. In
the complex protein-protein interaction (PPI), one possible technique to
estimate the information contained in the network is to measure entropy of the
network from which information contained in the network can be estimated Morzy
; Zenil . The entropy of a network $G(V,N)$ with $N$ possible states and
adjacency matrix $[U]_{N\times N}$, which is the amount of uncertainty one can
observe in accessing the states of the network Zenil , can be measured by,
$\displaystyle\mathcal{H}[U(G)]=-\sum_{i=1}^{N}p[U(x_{i})]lnp[U(x_{i})]$
Shannon . The maximum value of $\mathcal{H}$ of $U$, which have $N^{2}$
elements $\\{x_{i};i=1,2,...,N^{2}\\}$ and $x_{i}=0,1(binary)$, is given by,
$\mathcal{H}_{max}=log_{2}[N^{2}]$ from which one can estimate the information
contain in the network $G$ by, $I=\mathcal{H}_{max}-\mathcal{H}$ Kolmogorov .
There are various ways to calculate entropy of a network, namely, graph
entropy Lian  : define probability of a node $n$ to have inner links $e$ as
$p_{i}(n)=\frac{e}{|N(n)|}$ and probability of this node $n$ to have outer
links by $p_{o}(n)=1-p_{i}(n)$, then $\mathcal{H}(G)=\sum_{n\in V}h(n)$,
$h(n)=-p_{i}(n)lnp_{i}(n)-p_{o}(n)lnp_{o}(n)$, such that,
$I=\mathcal{H}_{max}-\mathcal{H}(G)$. Next method is to calculate signaling
entropy Teschendorff  for PPI network with each node represented by gene
expression data, where, the Pearson correlation value between any pair of
nodes $(i,j)$ $C_{ij}$ can be related to weight $w_{ij}$ assigned to the edge
by $w_{ij}=\frac{1}{2}[1+C_{ij}]$ from which one can obtain probability
distribution $\displaystyle p_{ij}=\frac{w_{ij}}{\sum_{j\in N(i)}w_{ij}}$ to
calculate entropy and information contained
$\displaystyle\mathcal{H}_{i}[G]=-\frac{1}{log(k_{i})}\sum_{i=1}^{N}[p_{ij}]log[p_{ij}]$
and $I=\mathcal{H}_{max}-\mathcal{H}$. On the other hand, simple network
centrality, namely, betweenness centrality (BC) metric can be used as a
significantly sensitive parameter to measure the capability of a node to
spread information in a complex network Newman2 . The reason could be due to
the fact that nodes having high centrality values can reach the other nodes in
the network on short paths Barthelemy  enhancing the speed of information
processing, and vice versa. This BC can then be correlated with the
connectivity (degree) $k$ in the network as a power law $C_{B}(k)\sim
k^{\epsilon}$ Goh . From this relation, one can estimate maximum BC as
$C_{B}^{max}\sim k_{max}^{\frac{\epsilon}{\gamma-1}}$, where, maximum
connectivity is a function of network size $N$ by, $k_{max}\sim
N^{\frac{1}{\gamma-1}}$, and $\gamma$ is the exponent of degree distribution
with $\gamma=[2,3]$ Barthelemy . Hence, centrality measurement metric can be
used to characterize information processing of a node in the network. However,
study on the changes in the topological properties in cancer systems in terms
of the variations in information processing in the whole networks and at
various levels of network organization at disease state are not studied
properly in detail.

Hubs in a complex network have interesting contrast roles depending on the
nature of the network, for example, if the network follows scale free then
hubs try to control and manipulate the whole network such that their removal
cause network breakdown Jeong1 , whereas, if the network obeys hierarchical
features, hubs are busy in correlating nodes in its own module/community as
well as with the communities such that their removal should not cause network
breakdown Ravasz ; Barabasi . These hubs are heterogeneous in nature and can
be classified into various types depending on their participation in the
network and roles Guimera . Hence, for a particular disease, it is quite
important to categorize the identified key/driver regulating genes/proteins to
understand their roles and potential in controlling the disease. Then, the key
regulators should be studied systematically in terms of their binding affinity
to various available drugs and medicinal compounds for possible drug
discovery.

In this work, we address structural behavior of the OC network to understand
the information processing and organization of the network along with the
nature of the hubs with their classification using network theoretical
approach. The behavior of the constituent communities at various levels of
organization in order to understand how OC network works at fundamental level.
Then the changes in the gene expression levels of the key regulators of the OC
and at their stages of the cancer. Then, the key regulators are allowed to
interact with the FDA approved available drugs to identify drug which could be
used to prevent/cure OC disease.

Results
We now present our analysis of the ovarian cancer (OC) data we have mined for
study. Our main focus is three fold, first, we calculated topological
properties of the complex network we have constructed using experimentally
verified list of genes/proteins (Methods) to capture the changes in the
calculated topological properties at cancer state and their roles in the
network organization. Second, the roles of the identified novel key regulators
in the cancer state, specially, in signal transduction from information
theoretical approach. Third, search for novel natural compounds (drug (FDA
approved) or food nutrients) with respect to the identified key regulators.

Theory of fractal properties of the ovarian cancer network
We considered experimentally verified pre-processed 4818 genes with 16,320
interactions collected from the patients’ data mined from the ovarian cancer
dedicated and related database (see the workflow and Methods for detail
information of mining and pre-processing the data). The complex OC network is
analyzed using Grivan and Newman community finding method Girvan , and found
that the network has five levels of hierarchical organization of
communities/sub-communities till triangular motif level (Fig. 2), which are
the basic functional unit of network organization Milo  and quite abundant in
real networks Mangan ; Alon , and follows interesting systems level
hierarchical organization in the network Malik  as also supported by the
topological results of the network and its communities.

Let us represent the hierarchical scale free OC network by a graph of size $N$
$G(V,E)$, where, $V=\\{n_{i};i=1,2,...,N,n_{i}\in V\\}$ is the set of nodes,
and $E=\\{e_{ij};i,j=1,2,...,N,e_{ij}\in E\\}$ is the set of edges. The
topological properties of the network, namely, probability of degree
distribution $P(k)$, clustering coefficient $C(k)$, connectivity $C_{N}(k)$,
centrality measurement parameters, betweenness $C_{B}(k)$, closeness
$C_{C}(k)$ and eigenvector $C_{E}(k)$ centralities follow power law
properties,

$\displaystyle P(k)\sim k^{-\gamma};~{}~{}C(k)\sim
k^{-\alpha};~{}~{}C_{N}(k)\sim k^{-\beta}$ $\displaystyle C_{B}(k)\sim
k^{\epsilon};~{}~{}C_{C}(k)\sim k^{\kappa};~{}~{}C_{E}(k)\sim k^{\delta}$ (1)

where, the power law exponents have their own values. We also found that the
data of each topological distribution function of the primary network and the
constituent communities collapse into a single curve after interpolation of
the data following the MacKinnon and Kramer’s numerical procedure which
indicates that each topological distribution function ($U(k)\rightarrow
P/C/C_{N}/C_{B}/C_{C}/C_{E}$) from primary network to communities/sub-
communities down to motif level follow one parameter scaling law MacKinnon
given by,

$\displaystyle\frac{U}{k^{\lambda}}=H\left[\frac{\eta}{k^{\lambda}}\right]\rightarrow
constant$ (2)

Here, $\eta$ was calculated by fitting with the scaled data and found to be
approximately proportional to the minimum path length of the corresponding
network/community/sub-community, and we found that $H\rightarrow constant=b$.
This shows that $U(k)\sim bk^{\lambda}$, such that
$\mathcal{F}(k)=\frac{U}{b}\sim k^{\lambda}$. If we take $b$ as the
normalization constant such that $b=\displaystyle\sum_{i}^{N_{j}}U(k_{i})$,
where, $N_{j}$ is the size of the jth community/sub-community with $j=0$
corresponds to primary network $N_{0}\rightarrow N$, then,
$0\leq\mathcal{F}(k)\leq 1$ and $\mathcal{F}$ behaves just like probability
distribution function, such that,
$\lambda=\\{-\gamma,-\alpha,-\beta,\epsilon,\kappa,\delta\\}$ corresponding to
$P(k),C(k),C_{N}(k),C_{B}(k),C_{C}(k),C_{E}(k)$ respectively. Now if $\hat{T}$
is an operator which coarse grain the system with a scale factor $\omega$,
then it is clearly seen that:
$\displaystyle\hat{T}\mathcal{F}(x)=\mathcal{F}(\omega x)=(\omega
x)^{\lambda}=\omega^{\lambda}\mathcal{F}(x)$ Calvert . If one consider the
property $\mathcal{F}(k)$ of the primary OC network and the same scaled
property $\mathcal{F}^{\prime}(k^{\prime})$ of any one of the communities/sub-
communities of the network, then one parameter scaling law (2) provide the
renormalization relation
$\mathcal{F}(k)dk=\mathcal{F}^{\prime}(k^{\prime})dk^{\prime}$: $k\rightarrow
k^{\prime}:k=\omega k^{\prime}$ which gives us the following renormalization
relation,

$\displaystyle\mathcal{F}^{\prime}(k)=\omega^{\lambda-1}\mathcal{F}(k)$ (3)

This means that the topological properties of the OC network remains the same
if the network is scale up or down by a scale factor $\omega$ which carries
the fractal properties in the network organization Calvert  indicating self-
organizing behavior in the OC network organization Kauffman1 .

Theorem 1: The communities which organize complex hierarchical OC network
exhibit self-similar or fractal topological properties.
Proof: Since the OC network $G(V,N)$ is hierarchical scale free topology, the
network can be decomposed into $M$ communities
$g_{i}(V_{i},E_{i});i=1,2,...,M;\forall V_{i}\in V,\forall E_{i}\in E$ with
random size and structures (one can use various community finding methods,
namely, Girvan and Newman method Girvan , Louvain method Blondel , constant
potts model method Traag  and other related methods Fortunato ). The
topological properties of these communities $\mathcal{F}_{i}$ follow one
parameter scaling behavior (2) with the primary OC network and obeys
renormalized scaling law (3). Generally, these communities can be dependent on
different scaling parameters $\omega_{i}$, such that,
$g_{i}(V_{i},E_{i})\rightarrow\mathcal{F}_{i}=\mathcal{F}_{i}(k_{i})$ with
$k_{i}=\omega_{i}k$. Then, one can write: $\mathcal{F}_{i}(\omega_{i}k)\sim
k^{\lambda}\mathcal{F}_{i}(\omega_{i})$ with the scaling $k\rightarrow
k_{i}:k=\omega_{i}k$. Now using renormalization relation,
$\mathcal{F}(k)dk=\mathcal{F}_{i}(\omega_{i}k)dk_{i}$, we have the following
scaling law,

$\displaystyle\frac{\mathcal{F}_{i}(\omega_{i}k)}{\mathcal{F}(k)}=\omega_{i}^{1-\lambda}$
$\displaystyle\frac{\mathcal{F}_{i}(\omega_{i}k)}{\mathcal{F}_{j}(\omega_{j}k)}=\frac{\mathcal{F}_{i}(\omega_{i})}{\mathcal{F}_{j}(\omega_{j})}=\left[\frac{\omega_{i}}{\omega_{j}}\right]^{1-\lambda}$
(4)

This scaling law indicates that the topological properties of the communities
of OC network have fractal properties with the primary network as well as
among the communities also following the scaling law in
${}^{\prime}k^{\prime}$ given by equation (Network medicine in ovarian cancer:
Topological properties to drug discovery). The scaling law still holds true
for the organization of smaller communities at various levels of organization
till motif level.

Numerical proof of the theorem can be done by the analysis of the topological
properties OC network from primary network down to the motif level (Fig. 2).
The behavior of $P(k)$ as the function of $k$ for different networks at
various levels of organization (primary, communities, sub-communities down to
motif level as shown in Fig. 2 b) are approximately similar as shown by the
nearly parallel fitted lines to the data points, where, the calculated values
of $\gamma$s for all networks are approximately the same (Fig. 2 a.). This
clearly indicates that all the $P(k)$ data are collapsed to a single curve
obeying the one parameter scaling behavior given by equation (2). Similar one
parameter scaling nature can be found in the behaviors of $C(k)$ and
$C_{N}(k)$ (Fig. 2 a middle and lower panels).

The one parameter scaling property can also be observed in the behaviors of
centrality measurement parameters ($C_{B}(k),~{}C_{C}(k)$ and $C_{E}(k)$)
where we discussed the behaviors of high degree nodes in the respective
networks (Fig. 2 c all panels). The reason is that signal processing in each
respective network is mainly regulated by high degree hubs Albert ; Barabasi .
However, there is significant roles of the low degree nodes in regulating the
network behavior which we will be discussing also in this work. Then the data
of the constituent networks fall in a single curve obeying one parameter
scaling law given by (2).

Lastly, we calculated network $connectance$ or $density$ Newman1  given by,
$\Omega=\frac{2E_{N}}{N(N-1)}\sim\frac{\bar{k}}{N}$, where, $E_{N}$ is the
total number of edges in the network of size $N$ and $\bar{k}$ is the mean
degree of the network. If the network is dense then
$\displaystyle\lim_{N\rightarrow\infty}\Omega\rightarrow constant$, otherwise,
the network is sparse, where,
$\displaystyle\lim_{N\rightarrow\infty}\Omega\rightarrow 0$. We calculated
$\Omega$ for all communities at various levels of organization (Fig. 2 b) and
found that all the data points collapsed to a single curve which obey power
law behavior, $\Omega(N)\sim N^{-1}$ with good goodness of fit. The behavior
can be seen in two ways, first, $\Omega$ rapidly decreases as $N$ increases
till $N\sim 100$ then becomes nearly steady state which approximately
converges to small value, but does not go to zero. This indicates that the
complex OC network is weakly compact as $N$ increases. Second, since the data
points beautifully falls in a single curve which obey power law behavior, the
wiring/rewiring of the edges among the nodes of the communities/sub-
communities is done proportionately to keep the self-similarity in the
topological properties of the network.

Theorem 2:Hierarchical scale free OC network does not obey centrality
lethality rule.
Proof: Communities at a certain level of organization are loosely connected to
minimize wiring energy cost to maximize information processing among them
Gallos , but within each community nodes are connected densely with large
within-community clustering coefficients and significantly small path lengths
Bassett  to allow each individual community function independently Sporns .
Hence, we assume that the topological properties of the communities are nearly
independent of each other and we consider the network $G(V,E)$ has $M$
communities at a certain level of organization, such that,

$\displaystyle
G(V,N)=\bigcup_{i=1}^{M}g_{i}(V_{i},E_{i}),~{}~{}~{}g_{i}(V_{i},E_{i})\cap
g_{j}(V_{j},E_{j})=\phi~{}~{}~{}(disjoint~{}sets)~{}~{}~{}\forall i,j\in G.$
(5)

Now, if we consider degree $k$ as random variable which is integrable in the
set $G(V,N)$, and since the the topological distribution function
$\mathcal{F}(k)$ has the property
$\displaystyle\sum_{i=1}^{M}\mathcal{F}_{i}(k_{i})=1$, one can assume that
this function behaves like probability distribution function. Taking
distribution function of equation (5) and applying theorem of addition of
probability Gardiner , we have,

$\displaystyle\mathcal{F}[G(V,E)]=\mathcal{F}\left[\bigcup_{i=1}^{M}g_{i}(V_{i},E_{i})\right]=\sum_{i=1}^{M}\mathcal{F}\left[g_{i}(V_{i},E_{i})\right]$
$\displaystyle\mathcal{F}(k)=\sum_{i=1}^{M}\mathcal{F}_{i}(k_{i})$ (6)

Now, if most important hub/hubs in the OC network is is/are removed, the most
probable changes the perturbation can make in the network is that the
community/communities $g_{c+1},g_{c+2},...,g_{c+n}$ in which the nodes present
it/them will be affected the most causing rewiring of the nodes in the network
Fortunato , and in the worst scenario the community or some of the communities
may collapse loosing all corresponding topological properties Heylighen .
Hence, even though the affected communities become breakdown due to the
perturbation, then, corresponding topological properties will be vanished:
$\sum_{j=c+1}^{c+n}\mathcal{F}(k_{j})\rightarrow 0$, the other remaining
$(M-n)$ communities will be survived because of their tight interaction
keeping themselves small worldness property Bassett ; Sporns , and they will
reorganize the OC network with the scaling transformation of the random
variable $k$ by $k\rightarrow k^{\prime}:k^{\prime}=\theta k$ and $k_{i}$ by
$k_{i}\rightarrow k_{i}^{\prime}:k_{i}^{\prime}=s_{i}k_{i}=s_{i}\omega_{i}k$
so that
$\mathcal{F}_{i}(k_{i}^{\prime})=\mathcal{F}_{i}(s_{i}\omega_{i}k_{i})=(s_{i}\omega_{i})^{1-\lambda}\mathcal{F}_{i}(k)$
as given below,

$\displaystyle\mathcal{F}(k^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{F}(\theta k)=\theta^{\lambda}\mathcal{F}(k)\noindent$
(7) $\displaystyle=$
$\displaystyle\sum_{j=1}^{M-n}\mathcal{F}_{j}(k_{j}^{\prime})=\sum_{j=1}^{M-n}(s_{j}\omega_{j})^{1-\lambda}\mathcal{F}_{j}(k)$
(8)

The above equation indicates that removing hub/hubs below a critical number of
hubs does not cause the collapse of the OC network even though few communities
are broken down due to the perturbation, but the network reorganizes with
modified topological properties. Hence, emergence of tightly bound nearly
independent communities/sub-communities are the main reason in keeping complex
network integration, stability, adaptation and management of external and
internal perturbations and most importantly self-organization of the network,
where, the activities of the leading hubs are well regulated by the emergent
functional communities/sub-communities.

Proof from the OC network data analysis is as follows. We carry out hubs
knock-out experiment by systematically removing the first 10-400 leading hubs
from the primary OC network, then studied the changes in the topological
properties of the resulting networks and compared with those of primary OC
network (Fig. 3 and Fig. 4). We found the following observations. The
systematic removal of the hubs from the primary OC network upto certain
critical hub number $N_{C}^{T}$, the fractal properties of the resulting
networks strictly kept preserved with change in the power of the topological
distribution function:
$\mathcal{F}(k)\rightarrow\mathcal{F}^{\prime}(k^{\prime}):$ $k\rightarrow
k^{\prime};~{}\lambda\rightarrow\lambda^{\prime}$. However, if the number of
hubs removed is larger than this critical value, $N_{C}\rangle N_{C}^{T}$ then
the $k^{\prime}$ of the resulting networks become random such that the
$\mathcal{F}$ become randomly varied. For example, the exponent of power law
behavior of $P(k)$ become $\gamma\rangle 3$ and $\gamma$ show random variation
(Fig. 3 left panel and inset figure). For the case $N_{C}\langle N_{C}^{T}$
and depending on the values $\gamma$ one can classify as : (a) hierarchical
nature for $\gamma\langle 2ln2$; (b) scale free nature for the regime
$2ln2\langle\gamma\langle 3$, and (c) random for $\gamma\rangle 3$. For
clustering coefficient after $N_{C}\rangle 80$ it is difficult to see the
behavior because the calculated clustering coefficient values for those
resulting networks from which large number of hubs are generally small and
nearly overlapped and fitting to those data becomes quite meaningless (Fig. 3
middle panel). The scenario is distinctly visible in the behavior of
neighborhood connectivity, where, the power law exponent $\beta\rightarrow 0$
for $N_{C}\rangle N_{C}^{T}$ otherwise finite even though there is quite
fluctuations in the values (Fig. 3 panel 4 and inset). In the case of
centrality measurement functions also the exponent of the power law
distributions are either goes to zero ($\epsilon,\kappa\rightarrow 0$) or
become random (for $\delta$) for $N_{C}\rangle N_{C}^{T}$ (Fig. 4 and insets).
The goodness of the fit to each data measured by $r^{2}$ value is also quite
reasonably good ($0.97\rangle r^{2}\rangle 0.7$). From these results we can
conclude the following behavior of $\mathcal{F}$,

$\displaystyle\frac{\mathcal{F}^{\prime}(k^{\prime})}{\mathcal{F}(k)}$
$\displaystyle=$
$\displaystyle\frac{k^{\prime\lambda^{\prime}}}{k^{\lambda}}~{}~{}~{}~{}~{}~{}~{}for~{}N_{C}\langle
N_{C}^{T}$ (9) $\displaystyle\rightarrow$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}for~{}N_{C}\rangle N_{C}^{T}$

The scale free network which is obtained here is categorized based on the
value of $\gamma$ ($2\langle\gamma 3$), however, the rewired network still
have strongly modular structure in the network, where, $C(k)$ and $C_{N}(k)$
still exhibit power law behaviors. This modular structure becomes vanishing as
$\gamma\rightarrow 3$, and then as $\gamma\rangle 3$ the topological
properties become randomly distributed.

Proposition: Self-similarity process in OC network is given by the following
scaling law:

$\displaystyle\frac{\mathcal{F}_{c}(k_{c})}{\mathcal{F}(k)}$
$\displaystyle\sim$ $\displaystyle\langle k_{c}\rangle^{-D_{k}};$ (10)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
1+\rho\left[\frac{d_{c}}{d_{k}}\right]$ (11)

where, $d_{c}$ and $d_{k}$ are network diameters in $c$th community, and
$\rho$ is proportionality constant.
Proof: The average diameter, which is the shortest distance between two nodes
in a network, can be taken as network metric and has the form in hierarchical
network Jung ; Chung ; Boccaletti  : $\displaystyle
d\sim\frac{ln(N)}{ln(\langle k\rangle)}$. Then, one can directly write as
$N\sim\langle k\rangle^{d}$, where, $N$ is the network size from which scaling
law of $c$th community of the network is given by,

$\displaystyle N_{c}\sim\langle k_{c}\rangle^{D_{c}};~{}~{}D_{c}\sim
f(d_{c})\sim f\left(\frac{l_{c}}{l_{0}}\right)$ (12)

where, $D_{c}$ is the fractal dimension of $c$th community which can be
related to the diameter of the $c$th community of the network. $l_{c}$ and
$l_{0}$ are shortest length between two nodes in $c$th community and
characteristics length respectively. Here, $\langle k_{c}\rangle$ is taken as
the scaling parameter to understand scaling law in complex OC network. Now
network mass of the $c$th community can be obtained as Song ,

$\displaystyle\langle M_{c}\rangle=\frac{N}{N_{c}}\sim\langle
k_{c}\rangle^{D_{c}}$ (13)

where, scaling of $k$ to $k_{c}$ with scaling factor $\omega[\langle
k_{c}\rangle]$ is given by renormalization theory given below Song ,
$k\rightarrow k_{c}:~{}~{}k_{c}=\omega[\langle
k_{c}\rangle]k;~{}~{}\omega[\langle k_{c}\rangle]\sim\langle
k_{c}\rangle^{-D_{k}}$. Renormalization theory allows to express the
distribution function $\mathcal{F}(k)$ in terms of the number of nodes having
’k’ links $n(k)$ in primary network with the number of nodes having $k_{c}$ in
$c$th community $n_{c}(k_{c})$ as,
$n(k)dk=N\mathcal{F}(k)=n_{c}(k_{c})dk_{c}=N_{c}\mathcal{F}(k_{c})dk_{c}$.
Putting the power law distribution of $\mathcal{F}(k)\sim k^{-\lambda}$ and
scaling law in $\omega$ and equation (13) we have,

$\displaystyle\frac{N}{N_{c}}\sim\omega^{1-\lambda}\sim\langle
k_{c}\rangle^{D_{c}}$ (14)

The fractal dimensions $D_{c}$ and $D_{k}$ given in equation (12) can be
expressed as: $D_{c}\sim f(d_{c})\sim\omega_{c}d_{c}$ and $D_{k}\sim
f(d_{k})\sim\omega_{k}d_{k}$. Now, by using scaling in $\omega$ and equation
(14), one can reach the following relation,

$\displaystyle\lambda=1+\rho\left[\frac{d_{c}}{d_{k}}\right];~{}~{}\rho=\frac{\omega_{c}}{\omega_{k}}$
(15)

The equations (12) and (15), we could relate the fractal dimension to the
diameter of the network and mean degree $\langle k\rangle$, which is dependent
on various network topological properties, can be taken as a scaling parameter
to understand fundamental laws of network properties and their organization.
Further, the distribution function $\mathcal{F}$ can be obtained as follows,

$\displaystyle\frac{\mathcal{F}_{c}(k_{c})}{\mathcal{F}(k)}\sim\frac{k_{c}^{-\lambda}}{k^{-\lambda}}=\omega[\langle
k_{c}\rangle]=\langle k_{c}\rangle^{-D_{k}}$ (16)

The above equation indicates that if one knows the mean value of degree of a
network and network diameter one can able to know the fractal properties of a
network.

Significant role of lower degree hubs at cancer state in OC network
One way to understand the role of hubs in regulating a complex network and its
constituent communities/sub-communities is to study the network behavior as a
function of degree ($k$) because $k$ relates directly with various important
topological distribution functions Barabasi ; Albert ; Newman1 . One of the
most important functions of the hubs involved in a complex network is the
capability of triggering information flow among the nodes directly or
importantly influencing the information flow among the nodes which are not
connected directly, and can be characterized by behavior of centrality
measures Freeman . The behavior of betweenness centrality $C_{B}$ as a
function of $k$ is two-fold in OC network (Fig. 4 left panel). First, the
primary network and the networks in which significant number of hubs are not
removed (number of highest degree hubs removed$\langle$150) $C_{B}$ follows
two behaviors, (a) low degree nodes follow power law with larger exponent
$\epsilon$: $C_{B}^{[s]}(k)\sim k^{\epsilon_{s}}$, whereas, high degree hubs
obey power law with smaller value of $\epsilon$ $C_{B}^{[h]}(k)\sim
k^{\epsilon_{h}}$, such that, $\epsilon_{h}\rangle\epsilon_{s}$ (Fig. 4 left
panel inset). This indicates that lower degree nodes can able to communicate
others in efficient manner with shorter paths which may be due to the
possibility of these small degree nodes may be segregated with smaller paths
Clark ; Newman2 . On the other hand, high degree nodes have longer paths as
compared to small degree nodes, which could be due to sparsely distribution of
high degree nodes buried inside the network and proper link needed to have by
each of them might be reduced significantly at cancer state. Now, one can have
two sets of nodes of the whole network, $V_{s}=\\{n_{i}:i\in N_{s}\\}$ for
small degree nodes, and $V_{h}=\\{n_{i}:i\in N_{h}\\}$ for high degree nodes
such that $N_{s}+N_{h}=N$ and we can assume $V_{s}\cap V_{h}\sim\phi$, so
that, $V=V_{s}\cup V_{h}$. It has been shown that for large complex
distribution of betweenness centrality follows power law: $P(C_{B})\sim
C_{B}^{-\theta}$ Barthelemy , hence, the joint probability distribution of
betweenness centrality for the whole network can be written as, $P(V_{s}\cap
V_{h})=P(V_{s})P(V_{h})$ and one can have $P(V_{s}\cap V_{h})\sim P(C_{B})$.
Now, we have,

$\displaystyle P(C_{B})$ $\displaystyle=$ $\displaystyle\int
dkP(k)\delta(C_{B}-k^{\epsilon_{s}})\times\int
dkP(k)\delta(C_{B}-k^{\epsilon_{h}})$ $\displaystyle\sim$ $\displaystyle\int
dkk^{\gamma}\delta(C_{B}-k^{\epsilon_{s}})\times\int
dkk^{\gamma}\delta(C_{B}-k^{\epsilon_{h}})$ $\displaystyle\sim$ $\displaystyle
C_{B}^{-[2+\left(\frac{1}{\epsilon_{s}}+\frac{1}{\epsilon_{h}}\right)(\gamma-1)]}$
$\displaystyle\theta$ $\displaystyle\sim$
$\displaystyle\left[1+\frac{1}{2}\left(\frac{1}{\epsilon_{s}}+\frac{1}{\epsilon_{h}}\right)(\gamma-1)\right]$
(17)

This equation (Network medicine in ovarian cancer: Topological properties to
drug discovery) shows the connection among the power law exponents $\theta$,
$\epsilon_{s}$, $\epsilon_{h}$ and $\gamma$. Now, as the number of high degree
hubs are removed $\epsilon_{s}\rightarrow\epsilon_{h}$, and at a certain
critical number of hubs removed $N_{c}^{[t]}~{}(\sim 200)$ where the two
fitted power law in $C_{B}$ merges $\epsilon_{s}=\epsilon_{h}$ one can
retrieve
$\displaystyle\epsilon_{s}=\epsilon_{h}=\epsilon=\frac{\gamma-1}{\theta-1}$
Barthelemy ; Vazquez . Further, for $N_{c}\rangle N_{c}^{[t]}$, we found that
$\epsilon$ decreases such that $\displaystyle\lim_{N_{c}\rightarrow
large}\epsilon\rightarrow 0$ which indicates that the path length becomes
constant indicating information flow becomes constant.

Similar behavior is found in the behavior of closeness centrality (CC) which
is the measure of distance between a node to all other nodes in the network
segregated with close shortest paths from it Freeman1 . The nature of
closeness centrality of high degree nodes as a function of degree exhibits
power law $C_{C}^{[h]}(k)\sim k^{\delta_{h}}$, but the exponent $\kappa_{h}$
has comparatively smaller value as compared to that of lower degree nodes,
$C_{C}^{[s]}(k)\sim k^{\delta_{s}}$, $\delta_{h}\rangle\delta_{s}$ (Fig. 4
right panel). Since CC indicates the measure of geographical distance to
spread the information from a node in the network Newman2 ; Freeman1 , the
rate of information transfer coordinated by high degree hubs is significantly
reduced as compared to those of low degree nodes at OC state. One of the key
reasons could be the direct drastic changes in the expression of some of the
driver genes (high degree hubs) in the genetic network at cancer state, where,
rest of the existing nodes (mainly low degree nodes) in the network are
indirectly influenced Malik ; Martinez . However, as the low degree nodes
generally tie with high degree hubs Freeman2 , the low degree nodes might know
the changes in the information processing of the high degree hubs in the
network and probably might act accordingly to keep efficiency of information
processing in the network. Further, as the number of high degree hubs removed
is increased $\delta_{s}\rightarrow\delta_{h}=\delta$ and also
$\displaystyle\lim_{N_{c}\rightarrow large}\delta\rightarrow 0$ indicating
information spread becomes constant.

The nature of eigenvector centrality (EC) is also quite similar to that of
betweenness centrality and obeys power law behavior, where, the exponent of
the power law function of the low degree nodes of the OC network,
$C_{E}^{[s]}(k)\sim k^{\kappa_{s}}$, is larger than that of high degree nodes,
$C_{E}^{[h]}(k)\sim k^{\kappa_{h}}$, i.e. $\kappa_{h}\rangle\kappa_{s}$ (Fig.
4 middle panel and inset). The importance of EC is that this network metric
not only reveals the influence of a node in the neighboring nodes but also the
score of the connected nodes Bonacich . Hence, the significant decrease in the
value of $\kappa_{h}$ of the high degree hubs indicates that the signal
propagation and influence of the hubs at various layers of neighbors of nodes
starting from each one of them connecting till the end of the network Spizziri
is quite reduced at cancer state. Whereas, the low degree nodes, which
received change in influence of the OC network, maintain faster rate where
$\kappa_{s}$ is large as compared to $\kappa_{s}$. As seen before in other
centrality measures we have: $\kappa_{s}\rightarrow\kappa_{h}=\kappa$ as
$\displaystyle\lim_{N_{c}\rightarrow large}\kappa\rightarrow 0$, such that,
$C_{E}(k)\rightarrow constant.$ at large $N_{c}$ limit.

Provincial hubs are prominent regulators in OC network
The behavior of neighborhood connectivity $C_{N}(k)$ (Fig. 3 right panel)
obeyed power law nature with negative exponent which is the signature of
disassortativity nature Pastor ; Newman3 . This disassortativity nature, which
is again supported by the rich-club coefficient $\phi$ and normalized rich-
club coefficient $\phi_{N}$ results (Fig. 5), reveals that there is no
evidence of rich-club formation of the high degree hubs Colizza . Since the
normalized rich-club coefficient $\phi_{N}\langle 1$ for the primary network,
the high degree hubs are sparsely distributed in the OC network topology and
this value of $\phi_{N}$ is still found to be smaller for those networks after
few hubs are knockout $N_{c}\rangle 10$ indicating the correlation among the
high degree hubs become decreased significantly as more number of highest rank
hubs are removed from the network van . This weakening in the correlation
among the rich degree hubs in the complex OC network reduces the number of
attractors of the nodes in the network and loosen the connectivity among the
low degree and peripheral nodes in the network causing drastic functional
diversity Senden . Hence, the dominance of controlling of the network by these
rich-degree hubs such signal processing, functional structuring and many other
regulating mechanisms will be less probable in OC network. Even though the
hubs having $k\geq 30$ have the tendency to tie tightly among themselves
(increasing features of $\phi$ and $\phi_{N}$ in Fig. 5), their controlling
dominance is weak ($\phi_{N}$). Since the OC network is hierarchical topology,
the loosely bound low degree nodes and moderate degree hubs to the high degree
hubs might have segregated at cancer state and organized forming diverse
loosely connected communities/subcommunities Colizza  to maintain optimal
efficiency of the system’s properties. Hence, the signaling behavior of high
degree hubs could be different from that of lower degree nodes
($\epsilon_{s}\rangle\epsilon_{h}$, $\kappa_{s}\rangle\kappa_{h}$ and
$\delta_{s}\rangle\delta_{h}$ in Fig. 4) in order to keep network organization
optimal even at cancer state.

Now, in order to identify the type of the hubs which take prominent
responsibility for OC network regulation we calculated participation ratio
$P_{i}$ of the nodes with respect to $Z_{i}$ score (see Methods) showing the
different types of hubs for different number of leading hubs removed (Fig. 6).
The analysis of the evolution of the different types of hubs in
$(P_{i},Z_{i})$-plane as a function of $N_{c}$ show that provincial hubs are
more prominent as compared to other types of hubs, namely, kinless and
connector hubs. For this we defined the probability of finding each type of
hubs $P_{H}$ as the ratio of the number of each type of hub $N_{u};~{}\forall
u\in\\{k,q,p\\}$ ($N_{k}\rightarrow number~{}of~{}kinless~{}hubs$,
$N_{q}\rightarrow number~{}of~{}connector~{}hubs$ and $N_{p}\rightarrow
number~{}of~{}provincial~{}hubs$) to the size of the network $N$ as given
below,

$\displaystyle
P_{H}(N_{c})=\frac{N_{u}}{N}\rightarrow\begin{cases}\frac{N_{k}}{N},&kinless~{}hubs\\\
\frac{N_{q}}{N},&connector~{}hubs\\\
\frac{N_{p}}{N},&provincial~{}hubs\end{cases}$ (18)

From the Fig. 7 we could see that probability of finding kinless hubs is quite
low as compared to other types of the hubs (connector and provincial hubs) and
could able to see when one removed $N_{c}=25,30$ and surprisingly there is no
signature of it for $N_{c}\rangle 30$ and $N_{c}\langle 25$. However,
probability of finding of connector and provincial hubs could be seen starting
from primary network and in the regime $0\leq N_{c}\leq 50$ connector hubs
abundance dominate that of provincial hubs in the network. This indicates that
even though first leading 50 hubs from the OC network are removed or their
regulatory functions are suppressed due to signal manipulation at cancer
state, these connector and provincial hubs take over main responsibilities to
keep network integrity, stability, regulation and preserve network topological
properties. As the OC severity increases (cancer state moves to stage I, II,
III etc.) the number of genes expressed/suppressed start increasing Schaner
most probably causing in increase in the number of manipulated high degree
hubs in the OC network Bast . In such situation cancer driven hub
manipulation, both connector and provincial hubs take major responsibility in
network regulation of silencing of number of hubs $N_{c}\leq 100$. If the
number of manipulated hubs is $N_{c}\rangle 100$ only provincial hubs take
main responsibility of network integrity and regulation.

The key regulators of OC network Malik : CD44, EPCAM, KRAS and MCAM are found
to be provincial hubs, and AKT1 falls in connector hub (Fig. 5b lowest panel).
All the key regulators do no change their hub identity in the knockout
experiment of $N_{C}=10$, except CD44, where, it changes from provincial to
connector as around $N_{C}=40$ hubs are knockout from the OC network. This
indicates that provincial and connector hubs are quite important in
understanding OC network regulation in maintaining network stability,
properties and self-organization when the network is perturbed externally and
internally by cancer signal.

Patterns of mRNA expressions of the key regulators show diverse activities
Now, we studied the activities of the key genes Malik  from the mRNA
expressions of the five key regulators using GEPIA (Gene Expression Profiling
Interactive Analysis) online server (see Methods) where p-value $P\rangle
0.01$. The mRNA expression levels of these key regulators show diverse
activities at OC state as compared to that of normal expressions (Fig. 6a). We
found that EPCAM was over expressed at ovarian cancer state, whereas, MCAM was
down regulated but CD44 was significantly up regulated as compared to that of
normal state. However, AKT1 and KRAS were slightly up regulated as compared to
that of normal state. This is quite strongly correlated to the observed
survival analysis of these key regulators Malik , where, some regulators
(AKT1, CD44, MCAM) act as cancer suppressor, whereas, some of them (EPCAM and
KRAS) work in the cancer progression. Hence, expressions AKT1, CD44, EPCAM and
KRAS can be correlated to increasing or decreasing the risk of disease
progression, and hence, they can be potential prognostics markers or drug
targets in OC carcinogenesis. Then, the mRNA expression levels of the key
regulators are calculated and clearly show that at stage-II all the key genes
unregulated maximally, and then slowly down regulated as the cancer stage
approaches from $II\rightarrow IV$ (Fig. 6b). Moreover, another important
aspect of these results is that the overall mRNA expression level EPCAM at any
OC state and it’s expression level at any stage of the OC is maximum, whereas,
those of KRAS are minimum. Further, the expression levels of EPCAM at any
stages (II-IV) of OC remain equally over expressed. Across all stages (I–III),
EPCAM was the most frequently expressed biomarker Patriarca . EPCAM could
therefore be an ideal tumor antigen candidate for detecting cancer cells.
Overexpression of EPCAM was connected to advanced stage of disease and poor
overall survival in some tumor types, suggesting EPCAM as a possible
prognostic marker. AKT activation is prevalent in high-grade, late-stage
serous ovarian carcinomas (6–9), according to immunohistochemical
investigations, and may thus play a role in promoting tumor development Yuan ;
Altomare ; Cristiano ; Kurose ; Phung .

The TIMER web tool Taiwen  analysis showed that the expression of AKT1, CD44,
EPCAM, KRAS and MCAM genes was significantly correlated with one or more
tumor-infiltrating immune subsets (Fig. 7). We found that the immune
infiltrates, namely, CD4+ T-cells show positive correlations with the
expressions of AKT1, CD44, MCAM and KRAS; macrophages have positive
correlation with KRAS and MCAM; and neutrophils with the expression of KRAS.
This indicates that the key regulators have the most significant correlation
with immune infiltrates. Important aspects of KRAS is that it’s expression has
positive correlation with maximal infiltrates (Purity, CD4+ T cell,
macrophage, neutrophil, dendritic cell). Hence, it is quite important to
notice that these key regulators involve in immune infiltrates and regulations
connecting to OC with immune system Yigit .

Network based drug repurposing and docking analyses
The identified novel key regulators of OC network using network theoretical
approach are shown to be potential drug targets to prevent/cure this disease
because of their high connectivity with the rest of the nodes (both high and
low degree) in the network and the ability to rapidly transfer information in
the network. Key regulator genes could be the most appropriate target in
complex biological network for drug detection because of their high connection
and capacity to rapidly transfer informationPrasad . Hence, we use these five
regulators for possible identification of available list of the FDA approved
drugs. Now a total of 219 drug-gene interactions for these key genes were
identified (Supplementary Table). We found that out of these available list of
drugs, Uprosertib, Progesterone, Solitomab and Regorafenib drugs showed
significant strong interactions with the key genes AKT1, CD44, EPCAM and KRAS
respectively having the highest score (Fig. 8A). Many of the drugs targeting
these key regulators can be used either individually or in combination.
Uprosertib is a reversible pan-AKT inhibitor that interacts with adenosine
triphosphate Pachl ; Dumble ; Tolcher . A recent study demonstrated that
ovulation, the release of hormones (e.g. progesterone and estrogen),
inflammatory factors and growth factors among others, promoted the migration
of intrauterine-injected malignant cells towards the ovarian stromal
compartment to form ovarian tumors Yang . Solitomab is an EPCAM antibody that
is being developed for the treatment of patients with multiple EPCAM-
expressing solid tumors Brischwein ; English . Because of its capacity to
engage resting, polyclonal, CD8+, and CD4+ T lymphocytes for very potent,
redirected destruction of target tumor cells, solitomab has exhibited
exceptional anticancer activity in preclinical ovarian tumor xenograft models
English . Regorafenib is an anti-angiogenic oral multikinase inhibitor that
has been approved for use in a variety of solid cancers Poklepovic .

We then did molecular docking analysis of the filtered drug compounds with the
AKT1, CD44, EPCAM, KRAS and MCAM proteins as shown in Fig. 8B. Based on their
binding affinities and visual inspection of docked complexes for their ability
to shape H-bonds and interactions key genes and ligands (Fig. 8B). Binding
affinities of AKT1, CD44, EPCAM and KRAS with drug Uprosertib (-7.723
kcal/mol), Progesterone (-3.715 kcal/mol), Solitomab (-6.877 kcal/mol) and
Regorafenib (-3.934) respectively, showed greatest affinity of binding among
these. The key protein (AKT1, CD44, EPCAM and KRAS) is bind with drug with
hydrogen bond and other non-polar bond with residues of key proteins (Fig.
3B).

Conclusion and discussion
Identification of driver genes in OC and understanding their roles are still
open questions and critical study in this field is necessary to prevent this
disease and also finding a possible drug target/s to cure this disease. The
study can be done from diverse directions of experimental techniques,
theoretical methods, data analysis etc with interdisciplinary perspective. We
mined experimentally verified genes, which involved in OC, from various
dedicated databases and analyzed that data from network theoretical approach
to understand how OC network is organized, what is the role of the hubs, what
could be the mechanism of signal manipulation in OC etc.

The OC network carries fractal properties which is reflected in the
topological properties of the network as power law distributions. If one goes
deeper inside the network through communities/subcommunities till basic
building block motif level (OC network has five hierarchical levels of
organization) one can observe self-similar power law distributions which could
be the origin of fractal nature in the OC network. These
communities/subcommunities can be think of as coarse grained functional
modules which can be scale down to basic functional units whose functional
properties are self-similar because the topological properties of these
functional modules at all levels of organization obey one parameter scaling
law where all the curves collapse to a single curve by interpolating the
curves. We proved it from both from data analysis and general fractal theory
approach Mandelbrot  and breakdown of any of the functional module caused by
removing by knocking out few leading hubs from the network do not cause OC
network breakdown. This property reveals that OC network does not follow
centrality lethality rule Jeong1  and self-organized.

Understanding the role/s of the hubs are important because driver genes are
generally one of these hubs and they are meant for potential drug target to
enable to eradicate OC disease. The hubs knockout experiment we performed
clearly shows us few important features in the OC network. First, removal of
significant number of hubs from the OC network allows the transition from
hierarchical network to scale free topology. Continued removal of hubs bring
changes from the scale free topology to random feature which is the signature
of network breakdown. This indicates the significant role of high degree hubs
in shaping the OC network topology as well as keeping network stability.
Second, one can observe the decrease in the rate of information processing of
the leading hubs which could be due to the manipulation of activity of hub
nodes as compared to the activity of lower degree hubs. This manipulation of
activities of high degree hubs at cancer state leads to decrease in the number
of ’attractors’ in the network structure causing segregation of lower degree
hubs with short paths to become more active. However, the nodes in the OC
network try to keep the network properties, stability and organization after
rewiring among them.

There are different types of hubs in OC network but surprisingly it is not the
high rank hubs but moderate rank ’provincial’ and ’connector’ hubs which are
the keepers and safeguarders of the OC network at cancer state when the cancer
signal spreads all over the entire network. There is no signature of formation
of ’rich-clubs’ by the high rank hubs in the OC network causing weakening the
ties of the lower degree hubs in the network periphery with the hubs affecting
in the functional diversity of the network. Hence, there is quite complex
regulatory mechanisms in network organization when the system is at cancer
state. Further, when the OC moves to severity condition i.e. stage I, II, III
etc which could be modeled by allowing the network to destabilize more high
degree hubs Schaner  the network regulation becomes more complicated. Hence,
we need to study further on the cancer data at various severity stages so that
one can detect OC at early stage for prevention. Another important aspect is
to study the proper analysis of these key genes and clinical trials for
possible drug discovery to eradicate this cancer type.

We then studied the gene expression patterns of these key regulators at OC
state and their behaviors at different stages of the OC. The analysis
indicates that EPCAM has maximum over expression overall at various stages of
the cancer state which indicates this gene behaves as cancer enhancer in the
OC progression Zheng . It is also found that CD44 is also highly expressed as
compared to normal state and could also be another important target gene Gao .
Similarly, other remaining regulators are also important for analysis in terms
of drug targets and other analysis. These regulators are also positively
correlated with immune infiltrates such as CD4+ T cells etc indicating the
correlation of OC with immune system. Although CD44 expression has been
observed during the progression of ovarian carcinoma, the questions of whether
high CD44 expression indicates a relatively favourable prognosis Kayastha  or
aggressive tumor behaviour and a poor prognosis Kayastha  and whether CD44 has
any prognostic significance Berner ; Saegusa ; Cannistra  have remained
unresolved. To better understand the significance of CD44 expression in
epithelial ovarian cancer, researchers used immunohistochemistry to examine a
large number of samples and examined the prognostic value of CD44 using
multivariate and univariate analyses Sillanpaa . The finding points CD44
playing a role in establishing treatment resistance by stimulating the
PI3K/Akt signalling pathway Liu . In preclinical models, an RNA-based
bispecific CD44-EPCAM aptamer was created and tested. By combining single CD44
and EPCAM aptamers with a double-stranded RNA adaptor, it can block CD44 and
EPCAM at the same time. In vitro and in vivo investigations, the bispecific
CD44-EPCAM aptamer was much more effective than either single CD44 or EPCAM
aptamer at suppressing cell proliferation or inducing apoptosis in ovarian
cancer cells Zheng1 . The findings show that the KRAS-variant is a new genetic
marker for OC risk, and that it could explain genetic risk in a number of
hereditary ovarian cancer and breast syndrome families, according to the
researchers Ratner . In an in vitro situation similar to the patient’s tumour,
proliferation is aided by MER-mediated ERK and AKT signalling, which enhances
phosphorylated nuclear ER. When MAPK activity is increased, inhibiting both
the Mutated, Estrogen Receptor, and ERK pathways is required to stop
proliferation, which may explain the use of aromatase inhibitors rather than
tamoxifen Kato . When there are significant genomic co-alterations, such as
KRAS or PI3K, it has been suggested that personalised combination therapy may
be a key to establishing tumur control Sicklick , especially when certain gene
product pathways, such as KRAS or PI3K, are activated KatoS . It could be a
useful indicator for predicting ovarian cancer metastasis, recurrence, and
drug resistance, as well as a diagnostic, therapeutic, and prognostic tool.
However, it’s surprising that the stronger key regulators expression is, the
earlier the clinical stage is, implying that it’s a protooncogene. However, in
the context of prognosis, it also functions as a tumour suppressor gene. This
issue will be the subject of our future research.

Further, network theoretical approach was used to identify potential drugs
from the list of FDA approved drugs which have high affinity to bind with the
key proteins and high score in molecular docking analysis. The identified
drugs could be useful for the prevention of this disease and need to be done
critical study on it both from theoretical and experimental perspectives. We
propose a systematic investigation on the identified key/driver genes and need
to do various clinical trials and experiments.

Methods
Flowchart of the method
The methods and algorithms we have used to study OC network are summarized in
the flowchart in the Fig. 1.

Construction of PPI network and Knockout of nodes
We chose six highly cited ovarian cancer databases, namely, COSMIC database,
Gene cards database, Ovarian kaleidoscope database, Dragon database of ovarian
cancer, Curated ovarian database and OC- Gene database, and mined 4818
experimentally verified genes which are involved in OC. We then curated the
identified list of the genes by using CGI code written in Perl and removed
gene duplication and involved aliases in the list. From this method we could
able to retrieve 660 genes out of 2000. This curated list of the genes is then
cleaned manually as well as using Cytoscape 3.7.1 plugin by mapping to UniProt
(January 2016), and we arrived at final list of 600 genes. Now following the
principle of one gene one protein idea we constructed OC primary network using
the curated 600 genes using GeneMANIA App Sara  by uploading the 600 genes in
Cytoscape 3.7.1 Markiel . The primary network so constructed is composed of
4818 with 16320 physical interactions among the involved nodes.

The knockout experiments of the high degree hubs were carried out by removing
the first 10-400 leading hubs along with their associated edges from the
primary OC network systematically and analyzed the topological properties of
the resulting networks.

Network topological properties
Consider a complex network of size $N$ defined by a graph $G(V,E)$, where,
$V=\\{n_{i};i=1,2,...,N;\forall n_{i}\in V\\}$ and
$E=\\{e_{ij};i,j=1,2,...,N;\forall e_{ij}\in E\\}$ are sets of nodes and edges
respectively. Since the PPI we constructed is undirected and unweighted
network, $G(V,E)$ can be constructed by defining an adjacency matrix
$[A]_{N\times N}$ with matrix elements, $A_{ij}=1$ if there is an edge between
’i’th and ’j’th nodes in the network, otherwise $A_{ij}=1$, and $A$ is
symmetric matrix Newman1 . We used $Cytoscape3.6.0$ with modules Network
analyzer and CytoNCA Shannon ; Doncheva ; Tang  and Igraph for calculating the
topological properties of the OC network. We employed Louvain community
detection algorithm Blondel  for digging communities/subcommunities at various
levels of organization embedded in OC network. The topological properties of a
complex network which are calculated for analysis are the following:

Degree ($k$): This network metric is the total number of links a node has in a
network. If $k_{i}$ is the degree of its node of the network, then it can be
obtained by,

$\displaystyle k_{i}=\sum_{j=1}^{N}A_{ij}$ (19)

This network metric characterize how much a node can influence other remaining
nodes in the network Newman1 . Larger the value of $k_{i}$ more influential
the its node is and vice versa.

Probability of degree distribution, $P(k)$: It indicates the measure of the
probability of finding nodes of degree $k$ in the network and can be
calculated as follows,

$\displaystyle P(k)=\frac{n_{k}}{n}$ (20)

where, $n_{k}$ is the total number of nodes with degree $k$ in the network.
This network metric can able to characterize topological features of different
types of networks, such as, scale free and hierarchical network follow power
law distribution with $k$, whereas, small world or random network obey Poisson
distribution with $k$ Ravasz ; Albert ; Barabasi .

Clustering coefficient (local) $C_{i}(k)$: It is a network metric that
characterize the ability of a node to have local internal links with its
nearest neighbors, which can be measured by by the ratio of the number of
linked pairs of neighbors of a node ($L_{i}$) to total number of possible
pairs of neighbors of that node Newman1 . It is given by,

$\displaystyle C_{i}(k_{i})=\frac{2L_{i}}{k_{i}(k_{i}-1)}$ (21)

Higher the value of $C_{i}$ local nodes are densely connected. However, higher
degree nodes generally have lower value of clustering coefficient Albert . For
hierarchical network $C_{i}$ can be conjectured by $C_{i}(k)\sim k^{\alpha}$
with $\alpha\leq 1$, whereas, for scale free, random and small world network
$C_{i}(k)\rightarrow constant$ Ravasz ; Albert ; Barabasi . One can also
calculate average clustering coefficient of the network by using,
$C(k)=\displaystyle\frac{1}{N}\sum_{i=1}^{N}C_{i}(k)$.

Neighborhood connectivity $C_{N}(k)$: This metric characterize the average
ties of a node with the nearest neighbours of degree $k$ in a network defined
as follows Pastor ,

$\displaystyle C_{N}(k)=\sum_{s}sP(s|k)$ (22)

where, $P(s|k)$ is conditional probability of the links of a node with $k$
connections to another node having $s$ connections. If the $C_{N}(k)$ exhibit
power law behavior $C_{N}(k)\sim k^{\pm\beta}$ with $\beta\leq 0.5$, then the
network falls to have in hierarchical features Pastor , where, $+\beta$
behavior is the signature of $assortativity$ property of the network, whereas,
$-\beta$ indicates $disassortativity$ nature of the network. Networks with
assortativity nature is the evidence of $rich-club$ formation of high degree
hubs in the network Colizza .

Rich-club coefficient ${\phi(k)}$: This metric is the measure of how well high
degree hubs establish connections among themselves to control the network
regulation, and can be defined as the ratio of the number of links connected
among the rich-club member hubs ($U_{\rangle k}$) to the maximum possible
links they ($M_{\rangle k}$) can have Colizza , as given below,

$\displaystyle{\phi(k)}=\frac{2U_{>k}}{M_{>k}(M_{>}k-1)}$ (23)

Now the normalized rich-club coefficient can then be written as follows
Colizza ,

$\displaystyle\phi_{norm}(k)=\frac{\phi(k)}{\phi_{rand}(k)}$ (24)

where, $\phi_{rand}(k)$ is the calculated rich-club coefficient of the
generated random networks with similar size and degree sequence. If the
network have signature of rich-club then $\phi_{norm}(k)\rangle 1$, otherwise
($\phi_{norm}(k)\langle 1$), there is rich-club formation.

Betweenness centrality $C_{B}$: It is the measure of the influence of a node
has to propagate an information in the other nodes of the network Newman1 ;
Borgatti ; Freeman3  which can be quantified as follows,

$\displaystyle C_{b}(v)=\sum_{i,j;i\neq j\neq k}\frac{d_{ij}(v)}{d_{ij}}$ (25)

where, $d_{ij}(v)$ denotes the number of geodesic paths from node $i$ to node
$j$ passing through node $v$ and $d_{ij}$ is the all possible number of
geodesic paths from node $i$ to node $j$. Hence, normalized betweenness
centrality $C_{B}$ is given by,

$\displaystyle C_{B}(v)=\frac{1}{M}C_{b}(v)$ (26)

where, $M$ denotes the number of node pairs, excluding $v$ Higher the value of
$C_{B}$ faster the information processing throughout the network and vice
versa.

Closeness centrality $C_{C}$: It can be regarded as the measure of how far an
information can be propagated from a certain node to the rest of the other
nodes in the network Bavelas ; Newman1 ; Borgatti . This metric of any node
$i$ can be calculated from the harmonic mean of the all possible geodesic
paths $d_{ij}$ linking to the node $i$,

$\displaystyle C_{C}(k)=\frac{N}{\sum_{j}d_{ij}}$ (27)

Hence, larger the value of $C_{C}$ farther the information can be reached, and
vice versa Newman1 .

Eigenvector centrality $C_{E}$: This network metric is the measure of strength
of influence of a node to the rest of the nodes in the network having diverse
influential capabilities Bonacich . The eigenvector centrality of a node $i$
in a network can be obtained,

$\displaystyle C_{E}(i)=\frac{1}{\lambda}\sum_{j=nn(i)}{v_{j}}$ (28)

where, $nn(i)$ represents the nearest neighbors of node $i$ in the network
with eigenvalue $\lambda$ and eigenvector $v_{i}$ in the eigenvalue equation,
$Av_{i}={\lambda}v_{i}$ where, $A$ is the network adjacency matrix. $C_{E}$
values of all nodes in a network are always positive $C_{E}\rangle 0$, and it
is considered to be the extension of degree centrality Newman1 .

Techniques to classify hubs
Classification of different types of hubs in a network can be done by the
following network metrics.

Z-score parameter ($Z_{i}$): This parameter is generally known as within
module Z-score parameter which measures how strongly a particular node in a
module is linked to the rest of the nodes in the module Guimera . The Z-score
of a node $i$ having $k_{i}$ links in certain module ”$c$” can be defined as,

$\displaystyle Z_{i}=\frac{k_{i}-\langle k\rangle_{c}}{\sigma_{k_{c}}}$ (29)

where, $\langle k\rangle_{c}$ and $\sigma_{k_{c}}$ are the expectation value
and standard deviation of degrees $k$ of all nodes in the module respectively.
This $Z-$score can able to distinguish hubs from non-hub nodes depending on
its value, namely, $modular-hubs$ if $Z_{i}\geq 2.5$ (nodes having large
number of links) and $non-hubs$ if $Z_{i}\leq 2.5$ (lower degree nodes)
Guimera .

Participation co-efficient ($P_{i}$): The local and global roles of the
identified modular hubs by $Z$-score can be further classified depending on
the participation of the hubs locally within their own modules and globally by
establishing links to other modules other than its own module. This
participation co-efficient of node $i$ in a network having $N_{M}$ modules,
which have degree $k_{i}$, can be defined as Guimera ,

$\displaystyle
P_{i}=1-\sum^{N_{M}}_{s=1}\left(\frac{k_{ic}}{k_{i}}\right)^{2}$ (30)

where, $k_{is}$ denotes the degree of node $i$ in $c$ module. If
$P_{i}\rightarrow 1$ then the node $i$ is uniformly linked to all other
modules, whereas, if $P_{i}\rightarrow 0$ the node links to the nodes within
its own module Guimera . Generally, nodes which have $P_{i}\rangle 0.7$ has
larger participation in other modules. Now, different types of hubs can be
classified depending on the values of $Z$-score and $P_{i}$ in
$(P_{i},Z_{i})$-plane as follows: 1. kinless hubs: Hubs in the range
$Z_{i}\geq 2.5$ & $P_{i}>0.70$, 2. connector hubs: in the area $Z_{i}\geq 2.5$
& $0.30<P_{i}\leq 0.75$, and 3. provincial hubs: in the area $Z_{i}\geq 2.5$ &
$P_{i}\leq 0.30$ in the $(P_{i},Z_{i})$ space Guimera .

Validation of the expression pattern of key genes
The GEPIA Tang1  for Cancer Genomics was used to investigate the expression
pattern of the key genes. Tumor or normal differential expression analysis,
profiling according to cancer subtypes or pathological stages, patient
survival analysis, similar gene finding, correlation analysis, and
dimensionality reduction analysis are all customizable functions in GEPIATang1
.

Correlation of gene expression with tumor-infiltrating immune cells:
Considering the importance of immune dysregulation in ovarian cancer, we
explored the correlation between AKT1, CD44, EPCAM, KRAS and MCAM mRNA
expression and tumor-infiltrating immune cells. The web tool TIMER
(https://cistrome.shinyapps.io/timer/) Taiwen  was used. Six tumor-
infiltrating immune subsets, neutrophils, macrophages, CD8+ T cells, CD4+
T-cells and including B cells were analysed.

Network-Based Drug Repurposing:
The DGIdb Cotto  database was screened out for discovering therapeutic targets
for key genes. The DGIdb was used to produce a similar drug-gene interaction
network. DGIdb is a database that can anticipate the physical and functional
interactions between key genes and drugs in a query. The interactions in the
DGIdb database come from five different sources: automatic high-throughput lab
experiment data, text mining, co-expression interaction data, genomic context
interaction prediction, and previous knowledge from other databases.

Protein structure preparation:
The X-ray structures of some key genes of ovarian cancer like AKT1, CD44,
EPCAM, KRAS and MCAM are available on protein data bank (PDB) Berman
corresponding to PDB ids 9S9W, 1POZ, 4MZV, 4EPT and 6LYN, respectively. Prior
to docking studies, the proteins were processed using Schrodinger Protein
Preparation Wizard during which missing hydrogens were added, hydrogen bonds
were optimized and water molecules were deleted Madhavi .

Ligands preparation:
Nutraceuticals are bioactive phytochemicals that deliver health benefits and
are relatively safe to be used for prevention and treatment against disease. A
large number of drugs are FDA approved pharmaceutical drugs. The chemical
structures of high score ligands from gene-drug interaction of key genes were
obtained from PubChem kimS . The chemical structures of ligands were prepared
with LigPrep module Schrodinger which generated diverse, accurate and energy
minimized conformations using OPLS-2005 force field Madhavi .

Docking studies using Glide:
Glide uses a range of filters and performs a thorough search of the
conformational, orientational and positional space for ligand in the binding
site of receptor. The receptor was represented by creating a cubic grid of
size 30x30x30 Å centred on the active site residues for each protein by
Schrodinger’s Receptor Grid Generation program Halgren . Further the ligands
were screened against each receptor using high throughput virtual screening
(HTVS) approach of Glide Halgren . The high ranking ligands were then
subjected to stringent screening via extra precision (XP) Halgren  method to
eliminate false positives and acquire precise binding modes of compounds.
Interaction analyses
Maestro was used to compute interactions between the docked protein-ligand
complexes. 2D diagrams were generated depicting hydrogen bonds and
hydrophobically interacting residues. Then PyMOL Seeliger  was used to
visualize and study the resultant receptor-ligand complexes.

Acknowledgments
KC and RKBS are financially supported by UPE-II, under sanction no. 101 and
DBT-COE. MZM financially supported by DHR, Indian Council of Medical Research
under Young Scientist Research Fellowship).

Author Contributions
RKBS conceived the model. KC, MZM and RKBS did the numerical experiment. KC,
MZM and RKBS prepared the figures of the numerical results, analyzed and
interpreted the simulation results. RKBS, KC and MZM wrote the manuscript. KC,
MZM, PS and RKBS involved in the study and reviewed the manuscript.

Additional Information
Competing financial interests: The authors declare no competing financial
interests.

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Figure 1: Schematic diagram of the workflow of the methods implemented in the
study of ovarian cancer network to drug discovery.

Figure 2: Properties of hierarchical organization of ovarian cancer network:
(a) Plots of $P(k),C(k)$ and $C_{N}(k)$ as a function of degree $k$ of
networks, primary to motif level of the communities which incorporate novel
key regulators and their corresponding collapse of the data points in a single
curve following one parameter scaling law. The same is true for all the
communities at various levels of organization which we did not show here. (b)
Hierarchical organization of the complex OC network at various levels of
organization. The calculated network connectance $\Omega$ as a function of
network size $N$ showing one parameter scaling behavior. (c) Similar plots and
behaviors of centrality parameters $C_{B}(k),C_{C}(k)$ and $C_{E}(k)$
indicating one parameter scaling nature.

Figure 3: Hubs knock-out induced transition of network properties: Topological
properties of the systems level organization of ovarian cancer network after
knock-out experiment by removing first top 150, 200, 250, 300, 350 and 400.
The behaviors of degree distributions ($P(k)$), clustering co-efficient
($C(k)$), and neighbourhood connectivity ($C_{N}(k)$) after knock-out
experiments. The insets show the changes in the exponent values by knock-out
experiments of leading hubs and their statistical $r^{2}$ values of fitting
procedure.

Figure 4: Hub knock-out induced properties of centrality parameters:
Properties of centrality measurements of the ovarian cancer network after
knock-out experiments by removing first top 150, 200, 250, 300, 350 and 400.
The behaviors of betweenness centrality ($C_{B}(k)$), closeness centrality
($C_{C}(k)$), and Eigenvector centrality ($C_{E}(k)$) after knock-out
experiments. The insets show the changes in the exponent values by knock-out
experiments of leading hubs and their statistical $r^{2}$ values of fitting
procedure.

Figure 5: Involvement of different types of hubs in OC network regulation: (a)
Analysis of the role of hubs by analyzing participation ratio of the ovarian
cancer network by plotting $Z_{i}$ as a function of $P_{i}$ in the knock-out
experiments of leading hubs $N_{C}=10-200$. (b) The plots indicating the
probability of finding different types of hubs (Kinless, Connector and
Provincial) as a function of $N_{C}$ (The number of leading hubs removed in
the knock-out experiments). Rich-club analysis of ovarian cancer network,
where, plots of $\phi$ as a function of degree $k$ are shown for knock-out
experiments removing the first $N_{C}=10,40,70,100$ leading hubs from the
primary ovarian cancer network. Location of key regulators of OC network in
$(P_{i},Z_{i})$-plane.

Figure 6: Gene expression patterns of key regulators using GEPIA database: (a)
Gene expression levels of key regulators (AKT1, CD44, EPCAM, KRAS and MCAM) of
OC network which compare between normal (black color) OC patients (red color).
(b) Plots of gene expression levels of the same key regulators of OC network
at various cancer stages (I-IV).

Figure 7: Correlation of the OC key regulators with various cancer-
infiltrating immune subsets: Plots of the gene expression levels of the OC key
regulators (AKT1, CD44, EPCAM, KRAS and MCAM) as a function of infiltration
levels for various cancer-infiltrating immune subsets via the TIMER database.

Figure 8: Identification of potential drug compounds which have optimal
binding affinity with the key regulators: (A) Interaction network of key
regulators with FDA-approved drugs. The green circle represents drugs, blue
color diamonds represents key regulators and red color drug shows highest
score. (b) Molecular docking of key regulators and drugs with highest score.
Uprosertib, Progesterone, Solitomab and Regorafenib drugs showed significant
strong interactions with the key genes AKT1, CD44, EPCAM and KRAS respectively
having the highest score.