# Structural characteristics in network control of molecular multiplex
networks

Cheng Yuan  Zu-Yu Qian  Shi-Ming Chen  Sen Nie<EMAIL_ADDRESS>School of
Electrical and Automation Engineering, East China Jiaotong University,
Nanchang, Jiangxi 330013, People’s Republic of China

###### Abstract

Numerous real-world systems can be naturally modeled as multilayer networks,
enabling an efficient way to characterize those complex systems. Much evidence
in the context of system biology indicated that the collections between
different molecular networks can dramatically impact the global network
functions. Here, we focus on the molecular multiplex networks coupled by the
transcriptional regulatory network (TRN) and protein-protein interaction (PPI)
network, exploring the controllability and energy requiring in these types of
molecular multiplex networks. We find that the driver nodes tend to avoid
essential or pathogen-related genes. Yet, imposing the external inputs to
these essential or pathogen-related genes can remarkably reduce the energy
cost, implying their crucial role in network control. Moreover, we find that
lower minimal driver nodes as well as energy requiring are associated with
disassortative coupling between TRN and PPI networks. Our findings in several
species provide comprehensive understanding of genes’ roles in biology and
network control.

††preprint: APS/123-QED

##  I Introduction

Networks are prevalent in exploring the phenomenons and principles of our
daily lives, e.g., traffic [1, 2], financial [3, 4], biological [5, 6, 7, 8]
and social systems [9, 10]. As the ultimate goal to explore these systems is
to drive them into desired states, numerous advances have been achieved
against network control [11, 12, 13]. Controllability as the first step of
controlling, quantifies if the system can be driven from any initial state to
any desired state within finite time with finite external inputs. Liu et al.
[11] creatively combined the structural controllability theory with complex
networks and proposed a method to determine the minimal number of inputs
(driver nodes) to fully control the directed networks. Exact controllability
is another framework to analyze the controllability of complex networks with
arbitrary structures and link weights [14]. Meanwhile, the minimal inputs [15,
16], optimal control strategy [17, 18], and structural characteristics of
network in controlling have been explored [19, 20, 21, 22]. Though the network
is completely controllable in theory, it could be difficult to control it in
practice due to the huge energy requiring. Consequently, much effort has been
expended on various mechanisms in determining the energy cost of network
control [23, 24, 25, 26, 27, 28, 29, 30, 31].

Functioning of many systems usually requires the coupling between different
types of networks [32, 33, 34, 35, 36]. Such as traffic systems are
constituted by road network, railway network, subway network and so on [37].
Communication systems include different on-line and offline social networks
[38, 39]. Biological networks also function as the consequences of complex
interactions between different molecular networks [40, 41, 42, 43]. For
instance, proteins in the protein-protein interaction network (PPI) are
translated from genes in the transcriptional regulatory network (TRN) [40].

Recently, Mahajan et al. [40] explored the interactions between TRN and PPI
networks of different species and revealed the impact of multiplex
architectures in network robustness. They found that the functionally
essential genes and proteins are situated in important parts of the multiplex
networks. Here, we examine the association between genes’ functional
characteristics and their roles in the network control. We show that imposing
external inputs to these essential or pathogen-related genes can efficiently
reduce the control energy, despite the minimal driver nodes set tends to avoid
them. Moreover, we find that negative correlation between TRN and PPI layer
can simultaneously decrease the number of driver nodes and energy requiring.

##  II Model

Here, transcriptional regulatory networks (TRN) represent the interactions
among transcription factors and their target genes. Edges in the TRN encode
direct interactions between a transcription factor and its target genes. In
protein-protein interaction networks (PPI), each node represents a protein and
undirected link denotes the physical or binding interactions. Coupling links
between layers represent the interaction between the genes in TRN and the
proteins in PPI, in which the proteins translated from genes could also
regulate other genes. The coupling between TRN and PPI layers is one-to-one
correspondence, i.e., a gene in TRN is collected to a corresponding protein in
PPI layer [40]. The Schematic diagram of the multiplex networks is shown in
Figure 1. We employ the datasets in Ref. [40], which are collected from 7
species:  _H. pylori_ [44, 45],  _M. tuberculosis_ [46, 47],  _E. coli_ [48,
49, 50],  _C. elegans_ [50, 51, 52],  _A. thaliana_ [50, 51, 53],  _M.
musculus_ [50, 51, 54], and  _H. sapiens_ [50, 51, 54]. Since TRN and PPI
networks have different coverage of the genome and proteome, only the genes
and proteins presented in both layers are considered in our analysis.

Formally, the dynamics of each node in TRN-PPI multiplex networks is described
by

$\dot{\bm{z}}(t)=\bm{f}(\bm{z},\bm{v},t)$ (1)

where $\bm{z}(t)=(z_{1}(t),z_{2}(t),\dotsc,z_{N_{1}+N_{2}}(t))^{T}$ is the
state of $N_{1}+N_{2}$ nodes at time $t$. $N_{1}$ and $N_{2}$ are the size of
TRN and PPI networks, respectively.
$\bm{f}(*)=[f_{1}(*),f_{2}(*),\dotsc,f_{N_{1}+N_{2}}(*)]^{T}$ captures the
nonlinear dynamics of each node, and
$\bm{v}(t)=[v_{1}(t),v_{2}(t),\dotsc,v_{M}(t)]^{T}$ is the external inputs
imposed on the multiplex networks. For simplicity, assuming the system is at a
fixed point $\bm{z}^{*}$, where $\bm{f}(\bm{z}^{*},\bm{v}^{*},t)=0$, and
$\bm{x}(t)=\bm{z}(t)-\bm{z}^{*}$, $\bm{u}(t)=\bm{v}(t)-\bm{v}^{*}$. Equation 1
can be linearized as

$\dot{\mathbf{x}}(t)={\left[\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{array}\right]}\mathbf{x}(t)+B\mathbf{u}(t)$ (2)

where $\left[\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{array}\right]\equiv{{\frac{\partial\bm{f}}{\partial\bm{z}}}\bigg{|}_{\bm{z}^{*},\bm{v}^{*}}}$
is the global adjacency matrix of the network and captures the interactions
between $N_{1}+N_{2}$ nodes. Whereas, $A_{11}$ ($A_{22}$) describes the
connections between nodes within TRN (PPI) networks and $A_{12}$ ($A_{21}$)
captures the intra-connections between TRN and PPI networks.
$B\equiv{{\frac{\partial\bm{f}}{\partial\bm{v}}}\bigg{|}_{\bm{z}^{*},\bm{v}^{*}}}$
represents how the external inputs are imposed on nodes.

Note that if the linear system 2 is locally controllable along a trajectory in
state space, then the corresponding nonlinear system 1 is also controllable
along the same trajectory. In addition, the linear control predictions are
consistent with control of nonlinear dynamics [5]. Hereafter, to apply the
controllability and control energy framework, we focus on the linear system 2.

Figure 1:  Schematic diagram of TRN-PPI multiplex networks with different
interlayer couplings. The nodes in two layers represent the genes and
proteins, respectively. The red nodes are the minimal driver nodes which are
imposed on external inputs. The black directed lines represent interactions
between transcription factor and its target genes in TRN layer, and the black
undirected lines represent the interactions between proteins in PPI layer. The
red dash lines represent the one-to-one interactions between genes and their
corresponding proteins. The interlayer coupling are different in (a) and (b),
thus the minimal driver nodes set are changed.

##  III Methods

###  III.1 Minimal driver nodes and control energy

A system is controllable if it can be driven from any initial state to any
desired state with finite external inputs within finite time. The nodes
imposed on external inputs are driver nodes, and the minimal number of driver
nodes measures the controllability of a network, defined as $N_{\text{D}}$,
and $n_{\text{D}}=\frac{N_{\text{D}}}{N}$. According to exact controllability
framework [14], $N_{\text{D}}$ can be calculated as

$N_{\text{D}}=\max_{i}\left(\mu\left(\lambda_{i}\right)\right)$ (3)

where $\lambda_{i}\left(i=1,2,\dotsc,N\right)$ represent the eigenvalues of
adjacency matrix, and $\mu\left(\lambda_{i}\right)$ is the geometric
multiplicity. Moreover, the minimal driver nodes set can be identified in $B$
to satisfy the equation

$rank[A-\lambda^{M}I_{\text{N}},B]=N$ (4)

where $\lambda^{M}$ refers to the eigenvalue according to the maximum
geometric multiplicity $\mu(\lambda^{M})$. It was noted that the rank of
matrix $[A-\lambda^{M}I_{\text{N}},B]$ is determined by the number of linearly
independent rows. By performing the elementary column transformation on the
matrix $A-\lambda^{M}I_{\text{N}}$, we can obtain the linearly dependent rows.
Therefore, the external inputs described by $B$ should be imposed on the rows
to eliminate all linear correlations to make the matrix
$[A-\lambda^{M}I_{\text{N}},B]$ full-ranked, and the corresponding minimal
driver nodes set can be identified [14].

On the basis of optimal control theory [55], the energy required to control
the system is
$E(t_{0},t_{f})=\int_{t_{0}}^{t_{f}}\|\mathbf{u}(t)\|^{2}\mathrm{d}t$. If the
initial state $\mathbf{x}_{0}=0$ at $t=0$, the minimal control energy is

$E(t_{f})=\mathbf{x}_{t_{f}}^{T}W^{-1}(0,t_{f})\mathbf{x}_{t_{f}}$ (5)

where $W(0,t_{f})=\int_{0}^{t_{f}}e^{At}BB^{T}e^{A^{T}t}\mathrm{d}t$ is the
Gramian matrix. As the control energy decays to a nonzero stationary value
with increasing time $t$, the control energy discussed here is
$E(t_{f}\rightarrow\infty)$, and the Gramian matrix is
$W(0,t_{f}\rightarrow\infty)$. We set the elements as
$A_{ii}=-(\delta+\sum_{j=1}^{N}A_{ij})$, where $\delta=0.25$ to ensure the
stability of the whole system.

###  III.2 Degree-degree correlation

Associativity between TRN and PPI layers is measured by the Pearson’s
correlation coefficient $\rho$ [56]

$\displaystyle\rho$ $\displaystyle=cor(k_{\text{out}},K)$ (6)
$\displaystyle={{\sum_{i=0}^{n}(k_{\text{out}}(i)-\overline{k}_{\text{out}})\sum_{i=0}^{n}(K(i)-\overline{K})}\over{\sqrt{\sum_{i=0}^{n}(k_{\text{out}}(i)-\overline{k}_{\text{out}})^{2}}\sqrt{\sum_{i=0}^{n}(K(i)-\overline{K})^{2}}}}$

where $k_{\text{out}}$ is the out-degree of node in the TRN network, $K$ is
the degree of node in the PPI network and $n$ is the number of nodes in each
layer.

###  III.3 Simulated annealing

To finely tune the degree-degree correlation coefficient $\rho$ of multiplex
networks, we adopt the simulated annealing algorithm [57]:

1\. Randomly shuffle gene labels in the TRN network, while keep protein labels
in the PPI network unchanged. Then calculate the absolute difference between
the shuffled and desired degree-degree correlations as
$\Delta^{P}=|cor_{k,K}^{P}-cor_{k,K}^{D}|$, where $cor_{k,K}^{P}$ is the
present correlation between nodes with $k_{\text{out}}$ degree in TRN and
nodes with $K$ degree in the PPI. $cor_{k,K}^{D}$ is the desired correlation.
Save the present gene labeling in the TRN network.

2\. Randomly select a subset of size $N$ (we use $N=10$) from the genes in the
TRN network, and shuffle their labels randomly. Save this modified labeling of
the TRN as a new sampling. Then define the new difference as
$\Delta^{*}=|cor_{k,K}^{*}-cor_{k,K}^{D}|$, where $cor_{k,K}^{*}$ is degree-
degree correlation of the present network after randomly shuffling $10$ genes
labels, and $cor_{k,K}^{D}$ is the desired correlation.

3\. Calculate the difference $\Delta=\Delta^{P}-\Delta^{*}$. If $\Delta\geq
0$, accept the labeling proposed in step 2 as the current labeling for TRN.
Otherwise, accept the labeling proposed in step 2 with probability
$e^{\Delta\over T}$, where $T=T_{0}e^{-\lambda L}$ is temperature, $L$ is the
iteration number and $\lambda$ is rate parameter. The parameters are set as
$T_{0}=1,000$, $\lambda=0.01$.

4\. If $\Delta^{P}$ is smaller than the defined value (here we set the defined
value as $0.01$) then stop, else repeat from the step 2.

##  IV Results

Figure 2:  Controllability and driver nodes distribution of TRN-PPI multiplex
networks. (a) The fraction of minimal driver nodes of seven species. (b)-(h)
are the fractions of minimal driver nodes in two layers for (b) _C. elegans_ ,
(c) _H. pylori_ , (d) _A. thaliana_ , (e) _M. musculus_ , (f) _E. coli_ , (g)
_M. tuberculosis_ , and (h) _H. sapiens_. We obtain 100 minimal driver nodes
sets, and calculate the average value of the driver nodes fractions in two
layers. P-value is calculated by t-test.

To reflect the controllability of these TRN-PPI multilayer molecular networks,
we firstly examine the minimal driver nodes $N_{\text{D}}$ to achieve fully
control of the network in each species (Figure 2(a)). The characteristics of
networks for each species is shown in Table 1. We find that overall those
networks display lower $n_{\text{D}}\ (\sim 0.15)$, indicating that we need to
independently control about $15\%$ nodes to fully control them. With the
smallest average degree, _H. pylori_ yields the highest $n_{\text{D}}$. This
is consistent with previous finding that sparse network is more difficult to
control. Then, we examine the distribution of driver nodes between two layers,
implying that these driver nodes do not have any particular preference between
two layers for most of species (Figure 2(b)-(h)). However, we find that the
fraction of driver nodes associated with PPI layer is significantly larger
than that of TRN layer for _H. sapiens_ (p-value$<$0.001). To understand such
difference, we calculate the average degree $\langle k\rangle$ of TRN and PPI
layer, respectively (Table 1), showing that the difference of average degree
between two layers is most obviously in _H. sapiens_ species. The density
difference between two layers leads to the uneven distribution of driver nodes
in TRN and PPI networks.

Table 1:  The characteristics of the TRN-PPI multiplex networks for seven species. Species | $N$ | $L_{\text{TRN}}$ | $L_{\text{PPI}}$ | $\langle k\rangle_{\text{TRN}}$ | $\langle k\rangle_{\text{PPI}}$
---|---|---|---|---|---
C. elegans | 628 | 821 | 285 | 2.62 | 1.82
H. pylori | 228 | 156 | 76 | 1.37 | 1.33
A. thaliana | 482 | 385 | 398 | 1.60 | 3.30
M. musculus | 920 | 1,269 | 518 | 2.76 | 2.25
E. coli | 434 | 147 | 154 | 0.68 | 1.42
M. tuberculosis | 1,542 | 958 | 946 | 1.24 | 2.45
H. sapiens | 3,172 | 4,838 | 4,280 | 3.05 | 5.40

The essential and pathogen-related genes are critical for survival and healthy
status of an organism, thus conveying particular topological characteristics
in the corresponding multiplex networks [40]. Hence, we examine whether those
essential or pathogen-related genes are associated with prominent role in the
network control. To achieve this, we adopt the gene categories in Ref. [40],
where essential genes for human species are collected from the Online GEne
Essentiality (OGEE) database [58, 59], and five human pathogens from a
publicly available database HPIDB 3.0 [60, 61]. Then, we compare the fractions
of essential or pathogen-related genes and non-essential or nonpathogen-
related genes are selected as the driver nodes, respectively. Figure 3
indicates that proportion of essential or pathogen-related genes selected as
driver nodes is significantly lower than that of non-essential or nonpathogen-
related genes (p-value$<$0.001), regardless of species. The fundamental role
of these essential or pathogen-related genes is usually reflected as the
higher degree in TRN layer. Yet, most of the driver nodes tend to be the lower
degree ones in control [11].

Figure 3:  Distribution of driver nodes in different gene categories. (a) _H.
sapiens_ , (b) Herpes, (c) Papillomaviruses, (d) HIV, (e) Yersinia, (f)
Dengue. We obtain 100 minimal driver nodes sets, and calculate the fraction of
essential or pathogen-related genes from driver nodes set in all essential or
pathogen-related genes (left, blue box), and non-essential or nonpathogen-
related genes from driver nodes set in all non-essential or nonpathogen-
related genes (right, yellow box), respectively, then get the average value.
P-value is calculated by t-test.

We further explore the impact of interlayer correlation on the controllability
of networks. Note that the interactions between TRN and PPI networks have
already been established by their biological relationships. The purpose of our
analysis is to reveal whether the underlying coupling pattern in reality can
be partially explained in a network control manner. Therefore, we finely tune
the correlation between TRN and PPI layer through simulated annealing
algorithm, finding that more driver nodes will be required as the network is
transited from disassortative to assortative (see Figure 4). We note that the
real interlayer correlation for the four species are mostly positive. We also
assess how degree correlation determines the energy requiring in network
control. We again finely tune the interlayer correlation by rewiring the
links. To eliminate the impact of different driver nodes sets, all nodes are
independently controlled. We find that control energy in terms of degree
correlation displays the similar trajectory as we observed in $n_{\text{D}}$
(in Figure 4), i.e., multiplex networks with disassortative coupling between
layers are easier to control than that of assortative coupling (see Figure 5).

Figure 4: The minimal number of driver nodes as functions of degree-degree
correlation of interlayers for TRN-PPI multiplex networks. (a) _E. coli_ , (b)
_M. musculus_ , (c) _A. thaliana_ , (d) _C. elegans_. The gray dash lines show
the real value of interlayers degree-degree correlation. Each data point is
the mean of 100 independent realizations. Figure 5:  Control energy as
functions of degree-degree correlation of interlayers for TRN-PPI multiplex
networks. (a) _M. musculus_ , (b) _A. thaliana_. The initial state is
$x_{0}=[0,0,\dotsc,0]^{T}$ and final state is $x_{t_{f}}=[1,1,\dotsc,1]^{T}$.
The number of driver nodes for two eukaryotes are $570$ and $200$,
respectively. The gray dash lines show the real value of interlayers degree-
degree correlation. Each data point is the mean of 100 independent
realizations.

Finally, we examine whether those essential genes can provide critical role in
the energy requiring for TRN-PPI multiplex network control. Subsequently, we
propose three driver nodes selection strategies with given $N_{\text{D}}$: (1)
All essential or pathogen-related genes, saying $N_{\text{E}}$ are selected as
driver nodes, then the remaining $N_{\text{D}}-N_{\text{E}}$ driver nodes are
selected randomly. (2) $N_{\text{D}}$ driver nodes are selected based on
descending order of nodes degree. (3) All $N_{\text{D}}$ driver nodes are
select at random. Interestingly, we find that the control energies of these
three strategies exhibit quite consistent implication that controlling those
essential or pathogen-related genes yields the lowest energy requiring. These
results imply the critical role of essential or pathogen-related genes in the
network control, despite the driver nodes tend to avoid them.

Figure 6:  Control energy as functions of the number of driver nodes by three
strategies. (a) _H. sapiens_. We choose all the $1,340$ essential genes as
driver nodes for _H. sapiens_ , and all the pathogen-related genes as driver
nodes for five different pathogens in (b)-(f). The number of pathogen-related
genes is $414$ for (b), $340$ for (c), $291$ for (d), $340$ for (e), and $116$
for (f).

##  V conclusion

Interactions between networks are ubiquitous in molecular networks. Here, we
focus on multiplex networks coupled with transcriptional regulatory network
(TRN) and protein-protein interaction (PPI) network, to explore the minimal
driver nodes and the energy requiring for fully control. The results indicate
there is no obvious preference for driver nodes distribution in two layers,
and the driver nodes are more likely to avoid the essential or pathogen-
related genes. Besides, the TRN-PPI networks with positive degree-degree
correlation of interlayer need more driver nodes and more energy to achieve
fully control. By comparing different strategies of selecting driver nodes, we
find the TRN-PPI networks requiring the lowest energy to reach the desired
state by driving essential or pathogen-related genes. Our work bridge the gap
between structural characteristics of molecular multiplex networks and the
network control. It will be helpful for understanding the essential genes’
function in biology and network control.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(No.61763013), The Natural Science Foundation of Jiangxi Province
(No.20202BABL212008), and the Jiangxi Provincial Postdoctoral Preferred
Project of China (No.2017KY37).

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