# Ambiguity in mana and magic definition and knot states

S. Mironovabcd, An. Morozovecd e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the Mana and Magic for quantum states. They have a standard
definition through the Clifford group, which is finite and thus classically
computable. We introduce a modified Mana and Magic, which keep their main
property of classical computability, while making other states classically
computable. We also apply these new definitions to the studies of knot states
of 2-strand knots.

a INR RAS, Moscow, 117312, Russia
b ITMP, MSU, Moscow, 119991, Russia
c MIPT, Dolgoprudny, 141701, Russia
d Kurchatov Institute, Moscow, 123182, Russia
e IITP RAS, Moscow 127994, Russia

##  1 Introduction

In recent years papers appeared which discuss magic and mana properties for
different models, such as CFT, knot theory and others [1, 2, 3]. These
quantities characterize how far is the certain quantum mechanical state
defined for such a model from the element of the Clifford group [4]. According
to the Gottesmann-Knill theorem [5], Clifford group elements can be
effectively modeled on a classical computer. Thus it is claimed that “magic”
is in effect a non-classicality of a certain state, and mana measures this
non-classicality. These properties can be important if one discusses these
properties in relation to the quantum computations.

The Gottesman-Knill theorem is based on a fact that Clifford group is a finite
subgroup of studied group $G$, which is a tensor product of several $SU(N)$’s.
However, it is not the only finite subgroup. One can define infinitely many
such subgroups for the same group $G$. Among these the defining property of
the Clifford group is its connection to the sigma matrices. From the point of
view of quantum computing there is no need to demand this. Thus depending on
the set of problems one wants to present to quantum computer, mana can be
defined differently. Our claim is that mana is in fact a relative rather than
absolute property.

In the present paper we will present how the Clifford group is usually defined
and how it can be modified to get other finite subgroups. We will apply this
new mana definition to studying knot states.

Knot theory is a widely studied subject with lots of relations to other
theories. Among others there are connections between knot theory and quantum
computations which provide both approaches to calculate knot polynomials using
quantum algorithms, as well as describing quantum algorithms as some knot
configuration in effective topological field theory [14]-[19]. This involves
calculating knots using unitary matrices through the Reshetikhin-Turaev
algorithms [6]-[13]. Specifically for some particular series of knots any
quantum algorithm can be described as consecutive approximations by a series
of knots [18, 19].

However, in the present paper we discuss different approach to the knot
theory. Mana and magic are properties of the quantum states (density matrices)
rather than unitary operations. There is a way to define quantum states,
corresponding to a knot [2], using ideas of topological field theories [20,
21]. Matrix elements of this density matrix are made from the knot polynomials
in special points. Thus classicality of such states provides us with some
information on how can these knot invariants be calculable on classical
computer.

Paper is organised as follows. In Chapter 2 we define Clifford group, which is
a finite subgroup of the $SU(N)$ group. In Chapter 3 we provide a definition
of mana as it is given in other papers on the subject, such as [1, 2, 3]. In
Chapter 4 we discuss ambiguity in mana definition and show how can the
definition be modified to give mana connected to a differen finite subgroup of
$SU(N)$. In Chapter 4 we define quantum mechanical states, describing
different knots, according to [20, 21, 2]. In Chapter 5 we study how mana
looks like for the knot states and how it can be changed by defining Mana
differently.

##  2 Clifford group

Clifford group was first defined by D. Gottesman [4]. Let us take a system of
$d$ orthormal states $|k>$, $k=0\ldots d-1$. We take a pair of operators $z$
and $x$:

$\begin{array}[]{l}Z=\sum\limits_{k=0}^{d-1}\omega^{k}|k><k|,\ \ \ \
\omega=e^{\frac{2\pi i}{d}},\\\ \\\ X=\sum\limits_{k=0}^{d-1}|(k+1)\text{mod}\
d><k|.\end{array}$ (1)

Using these operators one could define generalized Pauli operators

$T_{aa^{\prime}}=\left\\{\begin{array}[]{l}i^{aa^{\prime}}Z^{a}X^{a^{\prime}},\
\ d=2,\\\ \\\ \omega^{-\bar{2}aa^{\prime}}Z^{a}X^{a^{\prime}},\ \
d>2,\end{array}\right.$ (2)

where $\bar{2}$ is a multiplicative inverse of $2$: $2\times\bar{2}\equiv
1\text{mod}\ d$. Form the generalized Pauli operators one can defined strings
of such operators – Pauli strings:

$T_{\mathbf{a}}=T_{a_{1}a^{\prime}_{1}}T_{a_{2}a^{\prime}_{2}}\ldots
T_{a_{n}a^{\prime}_{n}}.$ (3)

The Clifford group is defined as a set of unitary operators $U$ which
transform Pauli string into another Pauli string up to a phase:

$\mathcal{C}=\left\\{U\ \ :\ \
UT_{\vec{a}}U^{\dagger}=e^{i\phi}T_{\vec{b}}\right\\}.$ (4)

Clifford gates – elements of the Clifford group – act in the space of the size
$d^{n}$. There are three Clifford gates which generate the whole group. Two of
them act in one $d$-dimensional space, namely phase gate and Hadamard gate:

$\begin{array}[]{l}K=\left(\begin{array}[]{llllll}1\\\ &\omega\\\
&&\omega^{2}\\\ &&\ldots\\\ &&&\omega^{\frac{(d-1)(d-3)}{2}}\\\
&&&&\omega^{\frac{(d-1)(d-2)}{2}}\end{array}\right),\\\ \\\
H=\cfrac{1}{\sqrt{d}}\left(\begin{array}[]{llll}1&1&\ldots&1\\\
1&\omega&\ldots&\omega^{d-1}\\\ \ldots&\ldots&\ldots&\ldots\\\
1&\omega^{d-1}&\ldots&\omega^{(d-1)^{2}}\end{array}\right).\end{array}$ (5)

Also there is one operator which acts on a pair of $d$-dimensional spaces:

$S=\sum\limits_{ij}|i;i\oplus j><i;j|.$ (6)

In the $SU(2)$ case this operator is the CNOT-gate.

Interesting fact is that there is a finite (although quite large) number of
elements in the Clifford group. Due to this fact anything constructed from the
Clifford gates can be effectively simulated on a classical computer in
polynomial time. Interesting fact is that Clifford group includes the
entangling S-gate (CNOT). Thus non-classicality (magic) is in fact a different
parameter from the entanglement.

##  3 Mana

To measure the degree of magic of a certain state a quantity called mana was
introduced. It is defined through a phase space point operator $A_{\vec{a}}$:

$A_{\vec{a}}=d^{-n}T_{\vec{a}}\sum\limits_{\vec{b}}T_{\vec{b}}T^{\dagger}_{\vec{a}},$
(7)

where $T_{\vec{a}}$ are Pauli strings (3). using these operators one can
define discrete Wigner function $W_{\rho}(\vec{a})$ of a state with density
matrix $\rho$:

$W_{\rho}(\vec{a})=\frac{1}{d^{n}}\text{Tr}\ \rho A_{\vec{a}}.$ (8)

Phase space point operators form a complete orthonormal basis in the space of
$d^{n}\times d^{n}$ matrices, thus density matrix can be constructed from the
Wigner functions:

$\rho=\sum\limits_{\vec{a}}W_{\rho}(\vec{a})A_{\vec{a}}.$ (9)

This means that for any physical state when $\text{Tr}\ \rho=1$, set of Wigner
functions satisfy $\sum W_{\rho}(\vec{a})=1$. Mana is defined as a logarithm
of negativity of the set of Wigner functions:

$M(\rho)=\log\sum\limits_{\vec{a}}|W_{\rho}(\vec{a})|.$ (10)

Mana possess several interesting properties which describes why this
definition was chosen. First, it can be shown that mana is equal to zero if
and only if the density matrix $\rho$ is made from Clifford gates. Second,
mana is additive:

$M(\rho_{a}\otimes\rho_{b})=M(\rho_{a})+M(\rho_{b}).$ (11)

Third, mana is related to the second Renyi entropy $S_{2}$:

$M(\rho)\leq\frac{1}{2}(L\log d-S_{2}).$ (12)

##  4 Ambiguity

The main property of the Clifford group, related to the ease of classical
computation is its finiteness. However Clifford group is not the only finite
subgroup of the $U(d)^{\otimes n}$ group. From the Clifford group one can
easily construct other finite subgroups using just the rotation matrices.

Let us take as an example the case of one unitary group. Then instead of $X$
and $Z$ operators one can take

$\tilde{X}=WXW^{\dagger},\ \ \ \tilde{Z}=WZW^{\dagger},$ (13)

where $W$ is a matrix from the $SU(d)$ group. These provide another set of
matrices instead of the generalized Pauli matrices:

$\tilde{T}=WTW^{\dagger}.$ (14)

Instead of Pauli strings one will also get generalized strings which can be
rotated independently for each of the generalized Pauli matrices in the
string. These generalized Pauli strings will also be related by some finite
group:

$\tilde{\mathcal{C}}=\left\\{\tilde{U}\ \ :\ \
\tilde{U}\tilde{T}_{\vec{a}}\tilde{U}^{\dagger}=e^{i\phi}\tilde{T}_{\vec{b}}\right\\}.$
(15)

These matrices are related to the Clifford group matrices when there is only
one $U(d)$ group by rotation with $W$. Thus instead of Hadamard and Phase gate
this group will be generated by

$\tilde{K}=WKW^{\dagger},\ \ \ \tilde{H}=WHW^{\dagger}.$ (16)

When tensor product of several $U(d)$ groups is considered, each group can be
rotated independently of others. The $S$ operator from (6) should be modified
accordingly:

$\tilde{S}=\mathcal{W}_{2}S\mathcal{W}_{2}^{\dagger},$ (17)

where $\mathcal{W}_{n}=W_{1}\otimes W_{2}\otimes\ldots\otimes W_{n}$, with
matrices $W_{1}$ and $W_{2}$ acting on the corresponding pair of two
dimensional spaces.

By modifying the Clifford group, the definition of Mana also should be
modified. Namely phase space point operator instead of (7) will be equal to

$\tilde{A}_{\vec{a}}=\mathcal{W}A_{\vec{a}}\mathcal{W}^{\dagger},$ (18)

which also modifies the definition of Wigner function:

$\begin{array}[]{r}\tilde{W}_{\rho}(\vec{a})=\frac{1}{d^{n}}\text{Tr}\
\rho\tilde{A}_{\vec{a}}=\frac{1}{d^{n}}\text{Tr}\
\rho\mathcal{W}A_{\vec{a}}\mathcal{W}^{\dagger}=\\\ \\\
=\frac{1}{d^{n}}\text{Tr}\
\mathcal{W}^{\dagger}\rho\mathcal{W}A_{\vec{a}}=W_{\mathcal{W}^{\dagger}\rho\mathcal{W}}(\vec{a}).\end{array}$
(19)

##  5 Knot states

Let us discuss applications of the generalized mana definition to the knot
theory. To define mana for knots, one should first define some quantum states,
related to knots. This can be done as follows, based on the papers by Atiyah
[20] and Witten [21]. From the physics perspective knot theory is a three-
dimensional Chern-Simons theory with action

$S=\frac{k}{4\pi}\int d^{3}x\left(\mathcal{A}\wedge
d\mathcal{A}+\frac{2}{3}\mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A}\right),$
(20)

where $k$ is an integer, called level of the Chern-Simons theory. Knot
polynomials are equal to the Wilson-loop averages of the Chern-Simons theory:

$J^{\mathcal{K}}_{r}(q)=\left<\text{Tr}_{r}\oint\limits_{\mathcal{K}}d\vec{x}\vec{\mathcal{A}}\right>.$
(21)

Polynomial depends on the gauge group of the Chern-Simons theory, its level
$k$, representation $r$ of the gauge group and contour of integration
$\mathcal{K}$. In what follows for the sake of simplicity we will speak only
of the $SU(2)$ group, when knot polynomials are Jones polynomials. Contour of
integration is a closed curve in three-dimensional space or a knot. Finally
the answer for the Wilson-loop average happens to be a Laurent polynomial in
variable $q$, related to the level of the Chern-Simons theory:

$q=e^{\cfrac{\pi i}{k+2}}.$ (22)

Jones polynomials are in fact related to the representations of the quantum
group $U_{q}(sl(2))$, rather than $SU(2)$ group. When $k$ is an integer then
$q$ is a root of integer and then there is a finite number of highest-weight
representations of the $U_{q}(sl(2))$ quantum group. Namely for even $k$ there
are only representations, corresponding to the spin $m/2$ for $m\leq k$ [22].
This allows us to define a basis of representations for the knot states.
Namely, state function can be defined as follows:

$|\mathcal{K}>=\sum\limits_{j=0}^{k-1}J_{j}|j>.$ (23)

Using such a wave function one can also define a density matrix:

$\rho_{\mathcal{K}}=|\mathcal{K}><\mathcal{K}|,$ (24)

and study mana for such states. In the present paper we will discuss only two-
strand knots, which polynomials can be calculated using the following formula
[11]:

$J^{T[2,n]}_{j}=\sum\limits_{i=0}^{j}\frac{q^{2j-2i+1}-q^{-2j-2i-1}}{q^{j+1}-q^{-j-1}}\left((-1)^{i}q^{i^{2}-2ji-i+j}\right)^{n},$
(25)

where $n$ is a number of twists in a two-strand braid. For odd $n$ the closure
of braid produces a knot, while for even $n$ the result is a link.

##  6 Mana for the Knot states

Mana for two strand knots is periodical in parameter $n$ with period being
equal to $k+2$, but besides that mana for many knots coincide. On the Fig.1
mana for $k=2$ is displayed. Knots appear only if $n$ are odd integers in
(25). However, using formula (25), one can extend this definition to the other
values of $n$, which gives graph on the Fig. 2. This is purely an analytical
continuation of the results for knots to an arbitrary $n$ by using (25).

Figure 1: 1\. Mana for two-strand knots for $k=2$ in Clifford group basis
Figure 2: 2\. Mana for two-strand knots for $k=2$ with continuation to
arbitrary real $n$ in Clifford group basis.

Similarly mana for $k=3$ (see Fig.3) and $k=4$ (see Fig.4) can be produced.

Figure 3: 3\. Mana for two-strand knots for $k=3$ in Clifford group basis.
Figure 4: 4\. Mana for two-strand knots for $k=4$ in Clifford group basis.

For $k=2$, as can be seen from Fig.1, there are two distinct values of mana
(and in fact only two different density matrices). By changing the basis we
calculate mana in, one can make for the other half of 2-strand knots to be
equal to zero, while making the mana for the other set to be nonzero. This in
fact can be done not only for $k=2$, but also for other cases. This means that
by changing the basis one can make different sets of knots states to be
effectively calculable on a classical computer.

Specific property of knot states is that they are pure, i.e. $\rho$ is a
tensor product of ket and bra vectors (24). The state for unknot ($n=1$ in
(25)) is defined by a vector $v=\frac{1}{\sqrt{d}}[1,1,..,1]$ and
$\rho=v^{\dagger}\times v$ for any $k$ and always has mana equals to zero. By
transitivity of unitary group it is always possible to rotate any other knot
state to the same vector $v=Uv_{1}$. This immediately gives rotated density
matrix or, in other words, new basis for Clifford group,
$\rho=Uv_{1}^{\dagger}\times v_{1}U^{\dagger}=v^{\dagger}\times v$ with zero
mana.

In $k=2$ case the only distinguished state besides unknot is trefoil knot
($n=3$ in (25)) with state vector $v_{1}=\frac{1}{\sqrt{3}}[1,-1,1]$. The
easies way to rotate it to the unknot state vector $v$ is by using unitary
matrix

$S=\left(\begin{array}[]{ccc}1\phantom{0}&0\phantom{0}\phantom{0}&0\\\
0\phantom{0}&0\phantom{0}\phantom{0}&1\\\
0\phantom{0}&-1\phantom{0}&0\end{array}\right)$ (26)

. Of course this matrix is not unique due to stability subgroup SU(2) and U(1)
unambiguity in definition of $v$ or $v_{1}$. But this freedom is not enough to
rotate both $v$ and $v_{1}$ to zero-mana states.

Figure 5: 5\. Mana for two-strand knots for $k=2$ in the basis, rotated by
matrix $S$ from (26). Figure 6: 6\. Mana for two-strand knots for $k=3$ in the
basis, rotated by matrix $S$ from (27).

The $k=3$ case can be analytically solved in a similar way, the rotation
matrix between $v$ and $v_{1}$ can be found for instance in a following way
(rounded answer)

$S=\left(\begin{array}[]{cccc}0.07+0.5i\phantom{0}&-0.7-0.5i&0&0\\\
0.7-0.5i\phantom{0}&0.07-0.5i\phantom{0}&0&0\\\
0&0&0.07-0.5i\phantom{0}&0.7-0.5i\\\
0&0&-0.7-0.5i\phantom{0}&0.07+0.5i\end{array}\right)$ (27)

Thus we illustrated for k=2 and k=3 that in different basis different knot
states can have zero mana but only separately. There is no basis in which two
or more different knot states have zero mana simultaneously. That in fact
shows that even if the absolute value of mana for a state is meaningless (it
can always be set to zero), the set of values of mana for different states has
an invariant property, they can not be set to zero together, thus are not
calculable on a classical computer.

##  7 Conclusion

In this paper we described how the definition of magic and mana can be
changed, while keeping its main property - finiteness of classical
calculations. This can be done by rotating either the basis of Clifford group,
or rotating the density matrix, which is in principle the same.

We applied these principles to the studies of mana of knot states. Knots have
deep connections with quantum calculations through Reshetikhin-Turaev
formalism, which relies on unitary matrices and thus becomes a natural task
for quantum computer. Another important connection is topological quantum
computer which in turn relies on knots as its basic algorithms.

We studied the mana of 2-strand knots and we also managed to find rotation
matrices which allows to change which knots have zero mana and thus are
classically calculable.

The goal of this paper was to make it clear that, while mana is an interesting
property of the quantum state, which should measure its “classicality”, its
definition is in fact very subjective. It relies on the specific finite
subgroup of the $SU(N)$ group. However one can define a different finite
subgroup which also changes the mana definition. This new mana also in fact
measures classicality, but in a different basis.

Clifford group is only distinguished by its relation to the $X$ and $Z$
operators (1), which has historical significance, but is not always connected
to the real quantum systems, used to build quantum computers. In fact for many
quantum computers there is a certain freedom to choose different basic
operations, quantum gates. For example for topological quantum computer
natural choice of operations are $\mathcal{R}$-matrices, which has no relation
whatsoever to the $X$ and $Z$ operators. Thus it is important to understand
which states are “classical” in relation to the exact choice of basic
operations.

We showed using the example of knot states that depending on the basis
different knot states can have zero mana in different bases. This in fact
means that mana is a relative rather than absolute property. Calculations on a
quantum computer (or on a classical reversible computer, which is its
classical counterpart) are made by using unitary matrices (or permutation
matrices in the classical case), which transform one state of the quantum
system into another. If these two states can have zero mana simultaneously in
some basis, then the corresponding calculations can be effectively made on a
classical computer. Thus mana definition should be chosen specifically for the
problem we are trying to study.

While in this paper we used knot states purely as an example to show that mana
can be defined differently, it is very interesting to study the dependence of
mana in different bases on the knot, as well as mana for the states, related
to other knot polynomials. This however still remains to be studied.

## Acknowledgements

We are grateful for very useful discussions to N. Kolganov.

This work was supported by Russian Science Foundation grant No 18-71-10073.

## References

  * [1] C.D. White, C. Cao, B. Swingle, _“Conformal field theories are magical”_ , Phys. Rev. B 103, 075145 (2021), arXiv:2007.01303
  * [2] J.R. Fliss, _“Knots, links, and long-range magic”_ , JHEP 04 (2021) 090, arXiv:2011.01962
  * [3] K. Goto, T. Nasaka, M. Nozaki, _“Chaos by Magic”_ , arXiv:2112.14593
  * [4] D. Gottesman, _“Theory of fault-tolerant quantum computation”_ , Physical Review A. 57 (1): 127–137, arXiv:quant-ph/9702029
  * [5] D. Gottesman, _“The Heisenberg Representation of Quantum Computers”_ , Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, eds. S. P. Corney, R. Delbourgo, and P. D. Jarvis, pp. 32-43 (Cambridge, MA, International Press, 1999), arXiv:quant-ph/9807006v1
  * [6] V.G. Turaev, _“The Yang-Baxter equation and invariants of links”_ , Invent.Math. 92 (1988) 527-553
  * [7] N.Yu. Reshetikhin, and V.G. Turaev, _“Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix”_ , Commun.Math.Phys. 127 (1990) 1-26
  * [8] N. Reshetikhin, V.G. Turaev, _“Invariants of three manifolds via link polynomials and quantum groups”_ , Invent.Math. 103 (1991) 547-597
  * [9] A. Morozov, and A. Smirnov, _“Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix”_ , Nucl.Phys. B835 (2010) 284-313,  arXiv:1001.2003
  * [10] A. Smirnov, _“Notes on Chern-Simons Theory in the Temporal Gauge”_ , The Subnuclear Series: Volume 47, The Most Unexpected at LHC and the Status of High Energy Frontier, pp. 489-498 (2011),  arXiv:0910.5011
  * [11] A. Mironov, A. Morozov and An. Morozov, _“Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid”_ , JHEP 03 (2012) 034,  arXiv:1112.2654
  * [12] A. Mironov, A. Morozov and An. Morozov, _“On Hopf-induced Deformation of Topological Locus”_ , Pis’ma v ZhETF, vol. 107, iss. 11, pp. 759 – 760,  arXiv:1804.10231
  * [13] An. Morozov and A. Sleptsov, _“New symmetries for the $U_{q}(sl_{N})$ 6-j symbols from the Eigenvalue conjecture ”_, Pis’ma v ZhETF, vol. 108, iss. 10, pp. 721 – 722,  arXiv:1905.01876
  * [14] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, _“Non-Abelian Anyons and Topological Quantum Computation”_ , Rev. Mod. Phys. 80, 1083 (2008), arXiv:0707.1889
  * [15] Yu. Makhlin, S. Backens, A. Shnirman, _“Two-qubit operation on Majorana qubits in ordinary-qubit chains”_ , Pis’ma v ZhETF, vol. 108, iss. 11, pp. 779 – 780,
  * [16] L. H. Kauffman, S. J. Lomonaco Jr., _“Braiding Operators are Universal Quantum Gates”_ , New Journal of Physics,4(2002) 73.1-18;6(2004) 134.1-40, quant-ph/0401090
  * [17] D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov, _“Towards topological quantum computer”_ , Nucl.Phys. B926 (2018) 491-508,  arXiv:1703.00431
  * [18] N.Kolganov and An. Morozov, _“On Hopf-induced Deformation of Topological Locus”_ , Pis’ma v ZhETF, vol. 111, iss. 9-10, pp. 623 – 624,  arXiv:2004.07764
  * [19] N. Kolganov, S. Mironov and An. Morozov, _“Large k topological quantum computer”_ ,  arXiv:2105.03980
  * [20] M. Atiyah, _“Topological quantum field theories”_ , Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques 68 (1988) no. 1, 175–186
  * [21] E. Witten, _“Quantum field theory and the Jones polynomial”_ , Commun. Math. Phys.121(1989) 351-399,
  * [22] B. Abdesselam, D. Arnaudon, A. Chakrabarti, _“Representations of $U_{q}(sl(N))$ at Roots of Unity”_, J. Phys. A: Math. Gen. 28 5495 (1995), arXiv:q-alg/9504006


††thanks: The authors are with the Department of Electrical Communication
Engineering, Indian Institute of Science, Bangalore, India.
<EMAIL_ADDRESS>
# Dr-COVID: Graph Neural Networks for SARS-CoV-2 Drug Repurposing

Siddhant Doshi and Sundeep Prabhakar Chepuri

###### Abstract

The _2019 novel coronavirus (SARS-CoV-2)_ pandemic has resulted in more than a
million deaths, high morbidities, and economic distress worldwide. There is an
urgent need to identify medications that would treat and prevent novel
diseases like the 2019 coronavirus disease (COVID-19). Drug repurposing is a
promising strategy to discover new medical indications of the existing
approved drugs due to several advantages in terms of the costs, safety
factors, and quick results compared to new drug design and discovery. In this
work, we explore computational data-driven methods for drug repurposing and
propose a dedicated graph neural network (GNN) based drug repurposing model,
called Dr-COVID. Although we analyze the predicted drugs in detail for
COVID-19, the model is generic and can be used for any novel diseases. We
construct a four-layered heterogeneous graph to model the complex interactions
between drugs, diseases, genes, and anatomies. We pose drug repurposing as a
link prediction problem. Specifically, we design an encoder based on the
scalable inceptive graph neural network (SIGN) to generate embeddings for all
the nodes in the four-layered graph and propose a quadratic norm scorer as a
decoder to predict treatment for a disease. We provide a detailed analysis of
the 150 potential drugs (such as _Dexamethasone_ , _Ivermectin_) predicted by
Dr-COVID for COVID-19 from different pharmacological classes (e.g.,
corticosteroids, antivirals, antiparasitic). Out of these 150 drugs, 46 drugs
are currently in clinical trials. Dr-COVID is evaluated in terms of its
prediction performance and its ability to rank the known treatment drugs for
diseases as high as possible. For a majority of the diseases, Dr-COVID ranks
the actual treatment drug in the top 15.

COVID-19; computational pharmacology; drug repurposing; graph neural network;
machine learning; SARS-CoV-2

##  I Introduction

The dreadful pandemic outbreak of the coronavirus disease 2019 (COVID-19) has
affected about 56 million people with more than a million deaths worldwide as
of November 2020. The June 2020 Global Economic Prospects [1] estimated a
$5.2$% downfall in the global gross domestic product (GDP) in 2020 that would
lead to the worst economic slowdown in history after the Second World War. The
disease affects mammals’ respiratory tract and shows symptoms similar to
pneumonia, causing mild to severe respiratory tract infections [2]. The
pathogen that causes COVID-19 belongs to the _Coronaviridae_ family, which is
a family of enveloped positive-strand RNA viruses that affect mammals, birds,
and amphibians. The name coronavirus (CoV) is derived because of the crown-
shaped spikes that project from their surface. Coronaviruses are majorly
grouped into four genera: _alphacoronavirus_ , _betacoronavirus_ ,
_deltacoronavirus_ , and _gammacoronavirus_. While _deltacoronaviruses_ and
_gammacoronaviruses_ infect birds, _alphacoronaviruses_ and
_betacoronaviruses_ infect mammals [3]. Out of the seven known strains of
human CoVs (HCoVs), the three _betacoronaviruses_ , namely, _middle east
respiratory syndrome coronavirus (MERS-CoV)_ , _severe acute respiratory
syndrome coronavirus (SARS-CoV)_ , and the _novel severe acute respiratory
syndrome coronavirus (SARS-CoV-2)_ produce severe symptoms. In the past two
decades, the world witnessed highly fatal _MERS-CoV_ and _SARS-CoV_ that led
to global epidemics with high mortality. Although the 2003 _SARS-CoV_ outbreak
was controlled, it infected 8098 individuals and resulted in 774 deaths. As of
November 2019, 2494 cases and 855 deaths were reported due to _MERS-CoV_ ,
with the majority in Saudi Arabia [3]. In December 2019, similar cases were
again reported in Wuhan City, China [4], wherein investigations confirmed it
to be the third novel CoV, i.e., _SARS-CoV-2_ , which is also referred to as
_HCoV-2019_ , _2019-nCoV_ , or colloquially simply as coronavirus [5]. _SARS-
CoV-2_ being highly contagious, on 30 January 2020, the World Health
Organization (WHO) declared it as a public emergency of international concern
warning all the countries with vulnerable health care systems [6].

The current treatment for COVID-19 is completely supportive and symptomatic as
there are no specific known medicines. Several research groups around the
world are trying to develop a vaccine that would prevent and treat _SARS-
CoV-2_. Looking at the current unpredictable trajectory of how the disease
spreads and the life cycle of the virus, there is an urgent need to develop
preventive strategies against it. Given this strict timeline, a more realistic
solution lies in drug repurposing or drug repositioning, which aims to
identify new medical indications of approved drugs. Drug repurposing offers
several advantages. It has a low risk of failure as the drug has already been
approved with less unknown harmful adverse effects. It reduces the time frame
for drug development as the drugs have passed all the pre-clinical trials and
safety norms. Finally, compared to the discovery of a new drug, drug
repurposing requires less economic investment and puts fewer lives of
volunteers (particularly kids) involved in clinical trials at risk [7]. Some
of the examples of repurposed drugs are Sildenafil, which was initially
developed as an antihypertensive drug was proved effective in treating
erectile dysfunction by Pfizer [7], and Rituximab that was originally used
against cancer was proved to be effective against rheumatoid arthritis [7], to
name a few. Even for COVID-19, drugs like Remdesivir (a drug for treating
Ebola virus disease), Chloroquine/Hydroxychloroquine (antimalarial drugs),
Dexamethasone (anti-inflammatory drugs) are being repurposed and are under
clinical trials as per the International Clinical Trials Registry Platform
(ICTRP), which is a common platform maintained by WHO to track the clinical
trial studies across the world.

Drug repurposing involves identifying potential drugs and monitoring their _in
vivo_ efficacy and potency against the disease. The most critical step in this
pipeline is identifying the right candidate drugs, for which experimental and
computational approaches are usually considered. To identify potential drugs
experimentally, a variety of chromatographic and spectroscopic techniques are
available for target-based drug discovery. Phenotype screening is used as an
alternative to target-based drug discovery when the identity of the specific
drug target and its role in the disease are not known [7]. Recently,
computational approaches are receiving attention due to the availability of
large biological data. Efficient ways to handle big data has opened up many
opportunities in the field of pharmacology. Zitnik, et al. [8] elaborates
several data-driven computational tools to integrate large volumes of
heterogeneous data and solve problems in pharmacology such as drug-target
interaction prediction (identify interactions between a drug and its target
genes), drug repurposing, and drug-drug interaction or side effect prediction,
to list a few. Hence this field is known as computational pharmacology. Many
standard machine learning (ML) and deep learning (DL) techniques have been
applied in computational pharmacology. Drug-drug interaction was formulated as
a binary classification problem and solved using ML techniques like random
forest, support vector machines (SVM), and naive bayes [9], and using DL
models like deep multi-layer perceptrons and recurrent neural networks, to
name a few. DL techniques often outperform standard ML techniques [10, 11].
However, these methods lack the ability to capture the structural information
in the data, specifically the connections between different biological
entities (e.g., interactions between drugs and genes or between drugs and
diseases). A natural and efficient way to represent such structural
information is to construct a graph with nodes representing entities like
drugs, genes, diseases, etc., and edges representing the complex interactions
between these entities. Graph neural networks (GNNs) capture the structural
information by accounting for the underlying graph structure while processing
the data. Decagon, a GNN-based model designed for predicting the side effects
of a pair of drugs has proved its capability by outperforming the non-graph
based machine learning models in terms of its prediction performance [12].
Similarly, drug repurposing has been studied using computational methods such
as signature matching methods, molecular docking, and network-based
approaches. Recently, network-based and machine learning approaches [13, 14,
15, 16, 17], and GNN based approaches [18] and [19] have been proposed for
drug repurposing.

In this work, we propose a GNN architecture for COVID-19 drug repurposing
called Dr-COVID, which is a dedicated model for drug repurposing. We formulate
our problem by constructing a four-layered heterogeneous graph comprising
drugs, genes, diseases, and anatomies. We then build a deep learning model to
predict the links between the drug and disease entities, where a link between
a drug-disease entity suggests that the drug treats the disease. Specifically,
Dr-COVID is based on the scalable inceptive graph neural network (SIGN)
architecture [20] for generating the node embeddings of the entities. We
propose a quadratic norm scoring function that rank orders the predicted
drugs. All the network information and node features are derived from the drug
repurposing knowledge graph (DRKG) [21]. DRKG is a biological knowledge graph
compiled using several databases, and comprises entities like drugs, diseases,
anatomies, etc., and their connections. We leverage their generic set of low-
dimensional embeddings that represent the graph nodes and edges in the
Euclidean space for training. We validate Dr-COVID’s performance on the known
drug-disease pairs. Although we present the results and analysis for COVID-19,
Dr-COVID is generic and is useful for any novel human diseases. From a list of
150 drugs predicted by Dr-COVID for _SARS-CoV-2_ , 46 drugs are currently in
clinical trials. For a majority of diseases with known treatment, the proposed
Dr-COVID model ranks the approved treatment drugs in the top 15, which
suggests the efficacy of the proposed drug repurposing model. As we use the
SIGN architecture that does many computations beforehand, Dr-COVID is
computationally efficient as compared to the other GNN-based methods [18, 19].
Specifically, in contrast to [18] we include additional entities such as
anatomies as the side information in our graph. This additional information
provides indirect interactions between the disease and gene entities. The norm
scorer we design captures correlations between the drug and disease pairs, and
as a consequence, the model predicts many more drugs (e.g., _Brexanolone_)
that are in clinical trials as compared to the existing GNN-based and network-
based drug repurposing models.

##  II Results and discussion

In this section, we present the drugs predicted by Dr-COVID for COVID-19
according to their pharmacological classifications, and elaborate on their
roles in treating the disease. We individually predict drugs for the 27
entities that specify the _SARS-CoV-2_ genome structure as identified by
Gordon et al. [22]. This genome structure includes structural proteins,
namely, envelope (_SARS-CoV2-E_), membrane (_SARS-CoV2-M_), nucleocapsid
(_SARS-CoV2-N_), surface (_SARS-CoV2-spike_) proteins, 15 non-structural
proteins (nsp), and open reading frames (orf) that encode the accessory
proteins. We also predict drugs for 6 diseases related to CoV, namely, _SARS-
CoV_ , _Avian infectious bronchitis virus (IBV)_ , _MERS-CoV_ , _CoV-229E_ ,
_CoV-NL63_ , and _Murine coronavirus (MHV)_. We choose the top 10 ranked
predicted drugs for all these disease targets, combine them, and present them
as a single list of 150 drugs (after removing the duplicate entries). We refer
to these 33 (i.e., 27 entities related to the _SARS-CoV-2_ genome structure
and 6 CoV diseases) as COVID-19 nodes. Out of these 150 drugs, 46 drugs are in
clinical trials in different phases. We provide the predicted scores of all
the drugs for all the COVID-19 nodes using Dr-COVID in our github repository.
The software to reproduce the results are available at:
https://github.com/siddhant-doshi/Dr-COVID

Fig 1 gives a heatmap indicating the ranks of these 150 drugs. It is a matrix
representation in which the drugs are listed on the vertical axis and COVID-19
nodes on the horizontal axis. All the 150 drugs are grouped based on their
first-level anatomical therapeutic chemical (ATC) codes as indicated on the
left side. A colored patch in the heatmap indicates the rank of a drug for a
disease. The darker the patch, the better is the rank, as indicated by the
rank bar on the right side. As can be seen, a major portion of the heatmap is
covered with dark patches as we only consider the top 10 ranked drugs. We can
infer from the heatmap that cardiovascular drugs (e.g., _Captopril_ ,
_Atenolol_) and anti-inflammatory drugs (e.g., _Celecoxib_ , _Prednisone_) are
ranked high for the _alphacoronaviruses_ , and a combination of antiparasitic
(e.g., _Ivermectin_), corticosteroids (e.g., _Prednisolone_ ,
_Dexamethasone_), antivirals (e.g., _Cidofovir_), and antineoplastic drugs
(e.g., _Methotrexate_ , _Sirolimus_) in the case of _betacoronaviruses_.

Figure 1: Rank heatmap of the predicted drugs with their corresponding ATC
labels.

Fig 2 lists the drugs predicted by Dr-COVID, grouped based on their first
level ATC codes such as antiparasitic (P), respiratory system (R), and so on,
whereas Fig 1 emphasizes the ranks of these predicted drugs for the COVID-19
nodes. The majority of the corticosteroids we predict belong to the
respiratory system (R) class, which has been the primary target for the
coronaviruses, as reflected by the symptoms. However, COVID-19 has a multi-
organ impact on the human body and is not limited to the respiratory system
[23]. Complications due to the cytokine storm with the effects of angiotensin
converting enzyme (ACE) have led to cardiac arrest, kidney failure, and liver
damage resulting in many deaths. For these reasons, we see drugs from various
ATC classes are being considered for clinical trials. Next, we discuss in
detail these pharmacological classifications of some of the predicted drugs.

Figure 2: Predicted drugs for COVID, categorized based on their ATC labels.

Anti-inflammatory (AI) agents: Inflammatory cytokine storms are prominently
evident in COVID-19 positive patients and timely anti-inflammation treatment
is required [24]. Pneumonia caused by the coronavirus results in a huge amount
of inflammatory cell infiltration leading to acute respiratory distress
syndrome (ARDS), causing many deaths [25, 26]. A wide range of anti-
inflammatory treatments including glucocorticoids, non-steroidal anti-
inflammatory drugs (NSAIDs), immunosuppressants, inflammatory cytokines
antagonists like tumor necrosis factor (TNF) inhibitors, Janus kinase (JAK)
inhibitors, and interleukin-1-receptor antagonist (IL-1RA) are being
considered for COVID-19. Our model predicts in the top 10, steroids like
_Dexamethasone_ , _Hydrocortisone_ , _Methylprednisolone_ ; NSAIDs like
_Ibuprofen_ , _Aspirin (Acetylsalicylic acid)_ ; immunosuppressants like
_Sirolimus_ and _Methotrexate_ ; IL-1RA _Anakinra_ ; CoX2 inhibitor
_Celecoxib_. These drugs are currently undergoing clinical trials. Some of the
other corticosteroids like _Betamethasone_ , _Prednisone_ , and TNF inhibitor
_Certolizumab pegol_ were also ranked high.

Antiviral and anti-parasitic agents: Dr-COVID predicts nucleotide analogue
antivirals like _Acyclovir_ , _Valaciclovir_ , _Cidofovir_ , and _Entecavir_
that have shown positive results in terminating the RNA synthesis catalyzed by
polymerases of coronaviruses [27]. _Ivermectin_ and _Nitazoxanide_ are used
against many parasite infestations and are also known to have antiviral
properties. _Mebendazole_ is another similar anti-parasitic drug that Dr-COVID
ranked high. One of the recent reports shows that _Ivermectin_ is an effective
inhibitor of the _SARS-CoV-2_ and many other positive single-stranded RNA
viruses. A 5000-fold reduction in the virus titer within 48 hours in cell
culture was obtained with a single treatment (5$\mu M$) of _Ivermectin_ [28].

Statins and ACE inhibitors/ beta-blockers/ calcium channel blockers: Statins
are lipid-lowering drugs that inhibit the cholesterol synthesis enzyme (also
known as HMG-CoA reductase), which also has anti-inflammatory properties.
There have been implications of lipid metabolism in the _SARS-CoV-2_
pathogenesis [29], due to which there are reports on including statins in the
line of treatment for COVID-19. Dr-COVID predicts _Atorvastatin_ ,
_Simvastatin_ , and _Rosuvastatin_ , where all the three drugs are currently
in clinical trials. On the contrary, some studies show that statins tend to
increase the cellular expression of ACE inhibitors [30], to which the _SARS-
CoV2-spike_ protein binds at the entry-level in humans [31]. Analyzing this
issue, an observational study by Zang et al. [32] reported a reduced mortality
rate in the patients treated with statins and no adverse effect was observed
by adding an ACE inhibitor drug also to the line of treatment. These ACE
inhibitors are cardiovascular drugs causing relaxation of blood vessels that
are primarily used to treat high blood pressure and heart failure. Beta-
adrenergic and calcium channel blockers are other similar functioning drugs
that lower blood pressure, are also currently considered to treat COVID-19.
Dr-COVID predicts _Captopril_ (ACE inhibitor), _Atenolol_ (beta-blocker) and
_Nifedipine_ (calcium channel blocker), which are currently in clinical
trials. Additionally, the list of predicted drugs includes _Spironolactone_
and _Hydrochlorothiazide_ , that help prevent our body from absorbing too much
salt and eventually lowering the blood pressure and avoiding cardiac failure.

Miscellaneous: Dr-COVID also predicts some of the pre-discovered vaccines such
as _Rubella virus vaccine_ , which is majorly considered for all the
healthcare workers, the _Yellow fever vaccine_ [33], and the _Ebola zaire
virus vaccine (rVSV-ZEBOV)_. Further, we also have _Mercaptopurine_ , an
antineoplastic agent that has been considered as a selective inhibitor of
_SARS-CoV_ [34] in the list of predicted drugs. Antidepressant _Brexanolone_
that is currently considered for patients on ventilator support due to ARDS,
vasodilators _Nitroglycerine_ and _Alprostadil_ , nutritional supplements like
_Riboflavin_ (Vitamin B2) [35], _Niacin_ , _Cholecalciferol_ (Vitamin D3), and
_Iron_ are some more top-ranked drugs. Interestingly, _Ephedra sinica root_ ,
a herb generally used to treat asthma and lung congestion, and an ingredient
of lung cleansing and detoxifying decoction (LCDD), which is a widely used
traditional Chinese medicine [36] is one of the drugs predicted in our list.

In essence, Dr-COVID predicts drugs for COVID-19 from different
pharmacological classes like the corticosteroids, antivirals, antiparasitic,
NSAIDs, and cardiovascular drugs, as the disease does not target particular
anatomy and impacts multiple organs in the human body.

##  III Dataset

In this section, we describe the dataset that we use to train and test Dr-
COVID for COVID-19 drug repurposing. We also describe how we model the data as
a multilayer graph to capture the underlying complex interactions between
different biological entities. We derive the required information from DRKG,
which is a comprehensive biological knowledge graph relating genes, drugs,
diseases, biological processes, side effects, and other eight more entities
useful for computational pharmacological tasks like drug repurposing, drug
discovery, and drug adverse effect prediction, to list a few. DRKG gathers all
this information from six databases, namely, Drugbank [37], Hetionet [38],
GNBR [39], STRING [40], IntAct [41], and DGIdb [42]. From DRKG, we consider
four entities that are relevant to the drug repurposing task. The four
entities are drugs (e.g., _Dexamethasone_ , _Sirolimus_), diseases (e.g.,
_Scabies_ , _Asthma_), anatomies (e.g., _Bronchus_ , _Trachea_), and genes
(e.g., _Gene ID: 8446_ , _Gene ID: 5529_). All the genes are referred with
their respective Entrez IDs throughout the paper. We extract the details about
these entities specifically from the Drugbank, Hetionet, and GNBR databases.
We form a four-layered heterogeneous graph with these four entities in each
layer as illustrated in Fig 3a. The four-layered graph is composed of 8070
drugs, 4166 diseases, 29848 genes, 400 anatomies, and a total of 1,417,624
links, which include all the inter-layer and intra-layer connections. Next, we
discuss the interactome that we consider for drug repurposing.

Figure 3: (a) Four-layered heterogeneous graph illustrating the inter-layer
and intra-layer connections. (b), (c) and (d) Subgraph centered around the
drugs _Dexamethasone_ , _Ivermectin_ and _Simvastatin_ , respectively.

Interactome: There are inter-layered connections between the four layers and
some have intra-layered connections. The inter-layered connections are of
different types. The drug-disease links indicate treatment or palliation,
i.e., a drug treats or has a relieving effect on a disease. For example,
interaction between _Ivermectin-Scabies_ (as seen in Fig 3b) and _Simvastatin-
Hyperlipidemia_ (as seen in Fig 3d) are of type treatment, whereas _Atropine-
Parkinson’s disease_ and _Diclofenac-Osteoarthritis_ are of type palliation.
The drug-gene and disease-gene links are the direct gene targets of the
compound and the disease, respectively. _Gene ID: 4306_ , _Gene ID: 387_ ,
_Gene ID: 1786_ are some of the targets of the drug _Dexamethasone_ (see Fig
3b) and _Gene ID: 5509_ , _Gene ID: 859_ are target genes of the disease
_Malaria_. Some of the genes targeted by the drug (e.g., _Dexamthasone_ ,
_Ivermectin_ , _Simvastatin_) as well as by the _SARS-CoV-2_ virus (referred
to as shared genes) are shown in Fig 3 (b,c and d). These common gene targets
between a drug and a disease are one of the reasons for the drug to be a
potential repurposing candidate against the disease. The disease-anatomy and
gene-anatomy connections indicate how the diseases affect the anatomies and
interactions between the genes and anatomies. For example, _Gene ID: 2771_ and
_Gene ID: 3156_ belong to the _cardiac ventricle_ anatomy (see Fig 3d);
disease _Schizophrenia_ affects multiple anatomies like the _central nervous
system (CNS)_ and _optic tract_.

There are also intra-layered connections. The drug-drug and disease-disease
connections show the similarity between a pair of drugs and diseases,
respectively. The gene-gene links describe the interaction between genes
(e.g., epistasis, complementation) and form the whole gene interactome
network. This comprehensive gene network serves as a backbone for our model,
wherein we predict the unknown links between drugs and new diseases like
COVID-19 as they are connected through genes and anatomies. The anatomy
information helps in drug predictions by focusing on the local interactions of
genes related to the same anatomy as the genes targeted by the disease. Some
examples of the intra-layered connections are _Simvastatin_ -_Lovastatin_ and
_Gene ID: 23649_ -_Gene ID: 8480_ as seen in Fig 3d. While all these
interactions reveal the true relations between the entities, we also randomly
sample the no-drug-disease links, which give us negative control in the
learning process. For example, there is no link between _Simvastatin-Scabies_
, i.e., _Simvastatin_ is not known to treat or suppress the effects of
_Scabies_. Including such negative control in the training process makes our
model accurate and reliable.

HCoV interactome network: To specialize the drug repurposing model Dr-COVID
for COVID-19, as discussed before, we consider the four known HCoVs, namely,
_SARS-CoV_ , _MERS-CoV_ , _CoV-229E_ and _CoV-NL63_ , and two non-human CoVs
namely _MHV_ , and _IBV_. We consider interactions of these disease nodes with
human genes. There are 129 links between these six disease nodes and the gene
nodes [21]. In addition, we consider all the 27 _SARS-CoV-2_ proteins
(including the structured proteins, nsp, and orf) and their 332 links
connecting the target human genes as given by Gordon et al. [22]. In other
words, there are only disease-gene interactions available for these COVID-19
nodes. With this available information, we train Dr-COVID to predict possible
drug connections for these COVID-19 nodes.

##  IV Methods and Models

In the last few years, deep learning has gained significant attention from a
variety of scientific disciplines due to its extraordinary successes in
solving many challenging tasks like data cleansing, mining, and
classification, mainly for images, speech, or text datasets. However, in many
applications, the structure underlying data is not always Euclidean. Some
examples include social networks, transportation networks, brain networks,
sensor networks, chemical molecules, protein-protein interactions, meshed
surfaces in computer graphics, and the drug repurposing network, as discussed
above, to list a few. For these applications, more recently, deep learning for
graph-structured data, also known as geometric deep learning (GDL) [43], is
receiving steady research attention. GDL aims at building neural network
architectures known as graph neural network (GNNs) to learn from graph-
structured data. GDL models are used to learn low-dimensional graph
representations or node embeddings by taking into account the nodal
connectivity information. These embeddings are then used to solve many graph
analysis tasks like node classification, graph classification, and link
prediction, to list a few. GNN architectures are developed using concepts from
spectral graph theory and generalize the traditional convolution operation in
the convolutional neural network (CNN) to the graph setting. In this section,
we describe the proposed Dr-COVID architecture for COVID-19 drug repurposing
and describe numerical experiments performed to evaluate our model.

Consider an undirected graph $\cal{G}=(\cal{V},\cal{E})$ with a set of
vertices $\cal{V}=$ {$v_{1},v_{2},\cdots,v_{N}$} and edges $e_{ij}\in\cal{E}$
denoting a connection between nodes $v_{i}$ and $v_{j}$. We represent a graph
$\cal{G}$ using the adjacency matrix $\mathbf{A}\in\mathbb{R}^{N\times N}$,
where the $(i,j)$th entry of $\mathbf{A}$ denoted by $a_{ij}$ is $1$ if there
exists an edge between nodes $v_{i}$ and $v_{j}$, and $zero$ otherwise. To
account for the non-uniformity in the degrees of the nodes, we use the
normalized adjacency matrix denoted by
$\tilde{\mathbf{A}}=\mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}$,
where $\mathbf{D}\in\mathbb{R}^{N\times N}$ is the diagonal degree matrix.
Each node in the graph is associated with its own feature vector (referred to
as input feature). Let us denote the input feature of node $i$ by
$\mathbf{x}_{i}^{(0)}\in\mathbb{R}^{d}$, which contains key information or
attributes of that node (e.g., individual drug side effects). Let
$\mathbf{X}^{(0)}\in\mathbb{R}^{N\times d}$ be the input feature matrix
associated with the $N$ nodes in the graph $\cal{G}$ obtained by stacking the
input features of all the nodes in $\cal{G}$. The new embeddings for a node is
generated by combining information from its neighboring nodes (e.g., diseases
or genes) to account for the local interactions. This process of combining
information and generating new representations for a node is done by a single
GNN block. If we stack $K$ such blocks, we can incorporate information for a
node from its $K$-hop neighbors (e.g., in Fig 3c, the drug _Ivermectin_ is a
$2$-hop neighbor of the anatomy _Lung_ and is connected via _Gene ID: 8614_).
Mathematically, this operation can be represented as

$\displaystyle\mathbf{X}^{(k+1)}=g_{k}(\bar{\mathbf{A}}\mathbf{X}^{(k)}\mathbf{W}_{k}),$
(1)

where $\mathbf{X}^{(k)}\in\mathbb{R}^{N\times d_{k}}$ represents the $k$th
layer embedding matrix and $d_{k}$ is the embedding dimension in the $k$th
layer. Here, $\bar{\mathbf{A}}=\mathbf{I}+\tilde{\mathbf{A}}$, where the
identity matrix $\mathbf{I}\in\mathbb{R}^{N\times N}$, is added to account for
the self-node embeddings, $\textbf{W}_{k}\in\mathbb{R}^{d_{k}\times d_{k+1}}$
is the learnable transformation matrix, and $g_{k}(\cdot)$ is the activation
function in the $k$th layer. There exist several GNN variants such as graph
convolutional networks (GCN) [44], GraphSAGE [45], graph attention networks
(GAT) [46] and scalable inception graph neural network (SIGN) [20], to name a
few. GCN is a vanilla flavored GNN based on Eq (1). GAT gives individual
attention to the neighboring nodes instead of treating every node equally. To
address the issue of scalability, GraphSAGE uses a neighbor sampling method,
wherein instead of taking the entire neighborhood, we randomly sample a subset
of neighbor nodes. SIGN takes a different approach to solve the scalability
issue and introduce a parallel architecture. The proposed Dr-COVID
architecture is based on the SIGN approach due to its computational
advantages. The predicted list of drugs from other GNNs are available in our
repository. Next, we describe the proposed Dr-COVID architecture.

###  IV.1 Dr-COVID architecture

The proposed GNN architecture for _SARS-CoV-2_ drug repurposing has two main
components, namely, the encoder and decoder. The encoder based on the SIGN
architecture generates the node embeddings of all the nodes in the four-layer
graph. The decoder scores a drug-disease pair based on the embeddings. The
encoder and decoder networks are trained in an end-to-end manner. Next, we
describe these two components of the Dr-COVID architecture, which is
illustrated in Fig 4.

Figure 4: Dr-CoV architecture.

Encoder: The Dr-COVID encoder is based on the SIGN architecture [20], which
provides low-dimensional node embeddings based on the input features and nodal
connectivity information. Recall that the matrix $\mathbf{A}$ is the adjacency
matrix of the four-layered graph $\cal{G}$ and $\tilde{\mathbf{A}}$ is the
normalized adjacency. SIGN uses linear diffusion operators represented using
matrices $\mathbf{F}_{r}$, $r=1,2,\cdots$, to perform message passing and
aggregate local information in the graph. By choosing
$\mathbf{F}_{r}=\tilde{\mathbf{A}}^{r}$ we can incorporate information for
node $v$ from its $r$-hop neighbors. Here, $\tilde{\mathbf{A}}^{r}$ denotes
the $r$th matrix power. To start the information exchange between the nodes,
we assume that each node has its own $d$ dimensional feature, which we collect
in the matrix $\mathbf{X}\in\mathbb{R}^{N\times d}$ to obtain the complete
input feature matrix associated with the nodes of $\cal{G}$. We can then
represent the encoder as

$\displaystyle\mathbf{Z}=\sigma_{1}\left\\{\left[\mathbf{X}\boldsymbol{\Theta}_{0}\,\|\,\mathbf{F}_{1}\mathbf{X}\boldsymbol{\Theta}_{1}\|\,\cdots\|\,\mathbf{F}_{r}\mathbf{X}\boldsymbol{\Theta}_{r}\right]\right\\}\quad\text{and}\quad\mathbf{Y}=\sigma_{2}\left\\{\mathbf{ZW}\right\\},$
(2)

where Y is the final node embedding matrix for the nodes in the graph
$\cal{G}$, and
{$\boldsymbol{\Theta}_{0},\cdots,\boldsymbol{\Theta}_{r},\mathbf{W}$} are the
learnable parameters. Here, $\|$ represents concatenation and
$\sigma_{1}\\{\cdot\\}$ and $\sigma_{2}\\{\cdot\\}$ are the nonlinear tanh and
leaky rectified linear unit (leaky ReLU) activation functions, respectively.
The matrix
$\mathbf{F}_{r}\mathbf{X}=\mathbf{D}^{-\frac{1}{2}}\mathbf{A}^{r}\mathbf{D}^{-\frac{1}{2}}\mathbf{X}$
captures information about the local interactions over $r$-hop neighbors. Fig
4 shows the encoder architecture. The main benefit of using SIGN over other
sequential models (e.g., GCN, GAT, GraphSAGE) is that the matrix product
$\textbf{F}_{r}\textbf{X}$ is independent of the learnable parameters
$\boldsymbol{\Theta}_{r}$. Thus, this matrix product can be pre-computed
before training the neural network model. Doing so reduces the computational
complexity without compromising the performance.

In our setting, we choose $r=2$, i.e., the low-dimensional node embeddings
have information from 2-hop neighbors. Choosing $r\geq 3$ is not useful for
drug repurposing, as we aim to capture the local information of the drug
targets such that a drug node embedding should retain information about its
target genes and the shared genes in its vicinity. For example, the $1$-hop
neighbors of _Dexamethasone_ as shown in Fig 3b, are the diseases it treats
(e.g., _Asthma_), and the drugs similar to _Dexamethasone_ (e.g.,
_Methylprednisolone_) and its target genes (e.g., _Gene ID: 8446_ , _Gene ID:
387_). The $2$-hop neighbors are the anatomies of the target genes (e.g.,
_Bronchus_) of _Dexamethasone_ , and the drugs that have similar effects on
the diseases (e.g., _Hydrocortisone_ and _Dexamethasone_ have similar effects
on _Asthma_). It is essential for the embedding related to _Dexamethasone_ to
retain this local information for the drug repurposing task, and not much
benefit is obtained by propagating more deeper in the network.

Decoder: For drug repurposing, we propose a score function that takes as input
the embeddings of the drugs and diseases and outputs a score based on which we
decide if a certain drug treats the disease. Fig 4 illustrates the proposed
decoder. The columns of the embedding matrix Y, contains the embeddings of all
the nodes in the four-layer graph, including the embeddings of the disease and
drug nodes. Let us denote the embeddings of the $i$th drug as ${\bf
y}_{c_{i}}\in\mathbb{R}^{l}$ and the embeddings of the $j$th disease as ${\bf
y}_{d_{j}}\in\mathbb{R}^{l}$. The proposed scoring function $f(\cdot)$ to
infer whether drug $c_{i}$ is a promising treatment for disease $d_{j}$ is
defined as

$\displaystyle s_{ij}=f({\bf y}_{c_{i}},{\bf y}_{d_{j}})=\sigma\left\\{{\bf
y}^{T}_{c_{i}}{\boldsymbol{\Phi}}{\bf y}_{d_{j}}\right\\},$ (3)

where $\sigma\\{\cdot\\}$ is the nonlinear sigmoid activation function and
${\boldsymbol{\Phi}}\in\mathbb{R}^{l\times l}$ is a learnable co-efficient
matrix. We interpret $s_{ij}$ as the probability that a link exists between
drug $c_{i}$ and disease $d_{j}$. The term ${\bf
y}^{T}_{c_{i}}{\boldsymbol{\Phi}}{\bf y}_{d_{j}}$ can be interpreted as a
measure of correlation (induced by ${\boldsymbol{\Phi}}$) between the disease
and drug node embeddings. We use $d=400$ and $l=250$ in our implementation.
The model is trained in a mini-batch setting in an end-to-end fashion using
stochastic gradient descent to minimize the weighted cross entropy loss, where
the loss function for the sample corresponding to the drug-disease pair
$(i,j)$ is given by

$\displaystyle\ell(s_{ij},z_{ij})=wz_{ij}\left(\log\left(\frac{1}{\sigma(s_{ij})}\right)\right)+\left(1-z_{ij}\right)\log\left(\frac{1}{1-\sigma(s_{ij})}\right),$
(4)

where $z_{ij}$ is the known training label associated with score $s_{ij}$ for
the drug-disease pair, $z_{ij}=1$ indicates that drug $i$ treats disease $j$
and otherwise when $z_{ij}=0$. Here, $w$ is the weight on the positive samples
that we choose to account for the class imbalance. As discussed in the Dataset
Section, we include the no-drug-disease links as negative control while
training our model. The number of no-drug-disease links is almost thirty times
the number of positive samples. To handle this class disparity, we explicitly
use a weight $w>0$ on the positive samples.

###  IV.2 Model evaluation

In this subsection, we evaluate Dr-COVID and discuss the choice of various
hyper-parameters. The drug repurposing via link prediction can be viewed as a
binary classification problem, wherein a positive class represents the
existence of a link between the input drug and disease, and otherwise for a
negative class. We have 6113 positive samples (drug-disease links) in our
dataset. To account for the negative class samples, we randomly choose 200,000
no-drug-disease links (i.e., there is no link between these drugs and
diseases). These links are then divided into the training and testing set with
a $90\%-10\%$ split. To use mini-batch stochastic gradient descent, we group
the training set in batches of size 512 and train them for 20 epochs. Due to
the significant class imbalance, we oversample the drug-disease links while
creating batches, thus maintaining the class ratio (ratio of the number of
negative samples to the number of positive samples) of $1.5$ in each batch.
The weight $w$ on the positives samples (mentioned in Eq (4)), is also chosen
to be the class imbalance ratio of each batch, i.e., we fix $w$ to be $1.5$.

We perform experiments on three sequential GNN encoder architectures, namely,
GCN [44], GraphSAGE [45], and GAT [46] for the drug repurposing task, which we
treat as a link prediction problem, and compare with the proposed Dr-COVID
architecture. Specifically, the SIGN encoder in Dr-COVID is replaced with GCN,
GraphSAGE, and GAT to evaluate the model performance. Two blocks of these
sequential models are stacked to maintain the consistency with $r=2$ of the
Dr-COVID architecture. We evaluate these models on the test set, which are
known treatments for diseases that are not shown to the model while training.
The model is evaluated based on two performance measures. Firstly, we report
the ability to classify the links correctly, i.e., to predict the known
treatments correctly for diseases in the test set. This is measured through
the receiver operating characteristic (ROC) curve of the true positive rate
(TPR) versus the false positive rates (FPR). Next, using the list of predicted
drugs for the diseases in the test set, we report that model’s ability to rank
the actual treatment drug as high as possible (the ranking is obtained by
ordering the scores in Eq (3)).

ROC curves show the performance of a binary classification model by varying
the threshold values used to classify the positive samples, which eventually
change the TPR and FPR. Fig 5 shows the ROC curves of different GNN models.
The area under ROC (AUROC), which lies in the interval [0,1] indicates the
separation ability of a binary classifier, where 1 indicates the best
performance, 0.5 means that the model is unable to discriminate between the
classes and 0 indicates a completely opposite behavior. We can see from Fig 5
that all the models have very similar AUROC values.

Figure 5: ROC Curves.

We also evaluate Dr-COVID in terms of ranks of the actual treatment drug in
the predicted list for a disease from the testing set, where the rank is
computed by rank ordering the scores as before. In addition, we compute the
network proximity scores [13] and rank order the drugs based on these scores
to compare with other GNN encoder models. These network proximity scores are a
measure of the shortest distance between drugs and diseases. They are computed
as

$\displaystyle
P_{ij}=\frac{1}{|\cal{C}|+|\cal{T}|}\left(\sum_{p\in\cal{C}}\min_{q\in\cal{T}}d(p,q)+\sum_{q\in\cal{T}}\min_{p\in\cal{C}}d(p,q)\right),$
(5)

where $P_{ij}$ is a proximity score of drug $c_{i}$ and disease $d_{j}$. Here,
$\cal{C}$ is the set of target genes of $c_{i}$, $\cal{T}$ is the set of
target genes of $d_{j}$, and $d(p,q)$ is the shortest distance between a gene
$p\in\cal{C}$ and a gene $q\in\cal{T}$ in the gene interactome. We convert
these into Z-scores using the permutation test as

$\displaystyle Z_{ij}=\frac{P_{ij}-\mu}{\omega},$ (6)

where $\mu$ is the mean proximity score of $c_{i}$ and $d_{j}$, which we
compute by randomly selecting subsets of genes with the same degree
distribution as that of $\cal{C}$ and $\cal{T}$ from the gene interactome, and
$\omega$ is the standard deviation of the scores generated in the permutation
test. Table 1 gives the rankings, which clearly show that the Dr-COVID results
in better ranks on the unseen diseases than the other GNN variants. Also,
compared to the network proximity measure, which is solely based on the gene
interactome, Dr-COVID performs better. We choose these drug-disease pairs for
evaluation as these links are not shown during the training. It is evident
that the diseases on which we evaluate are not confined to single anatomy
(e.g., _rectal neoplasms_ are associated to the _rectum_ anatomy, whereas
_pulmonary fibrosis_ is a _lung_ disease), nor do they require a similar
family of drugs for their treatment (e.g., _Fluorouracil_ is an antineoplastic
drug, and _Prednisone_ is an anti-inflammatory corticosteroid). Thus,
showcasing our model’s unbiased nature. For a majority of the diseases in the
test set Dr-COVID ranks the treatment drug in top 10 (as seen in Table 1). In
the case of _Leukemia_ (blood cancer), other antineoplastic drugs like
_Hydroxyurea_ and _Methotrexate_ are ranked high (in top $10$) and its known
treatment drug _Azacitidine_ is ranked $17$. We give more importance to the
ranking parameter as any drug predictor requires classifying and ranking the
correct drugs as high as possible. Considering this AUROC-ranking trade-off we
can see that Dr-COVID with SIGN encoder performs the best.

Disease | Treatment drug | Ranks
---|---|---
|  | Dr-CoV (SIGN) | SAGE | GCN | GAT | Network proximity
_Encephalitis_ | _Acyclovir_ | 10 | 35 | 35 | 295 | 5462
_Rectal neoplasms_ | _Fluorouracil_ | 9 | 421 | 16 | 231 | 2831
_Pulmonary fibrosis_ | _Prednisone_ | 5 | 3 | 10 | 9 | 2072
_Atrioventricular block_ | _Atropine_ | 6 | 79 | 8 | 14 | 4453
_Pellagra_ | _Niacin_ | 2 | 56 | 497 | 484 | Not computable
_Colic_ | _Hyoscyamine_ | 1 | 1 | 501 | 205 | Not computable
_Leukemia_ | _Azacitidine_ | 17 | 120 | 31 | 332 | 377

Table 1: Ranking Table. The Table gives the ranking performance of Dr-COVID
compared with other GNN variants and the network proximity measures. There are
no associated genes with some of the disease in our database, which makes it
impossible to compute the Z scores. These are indicated as “Not computable".
The best results are highlighted in bold font.

COVID-19 analysis: We perform a similar analysis and identify potential
candidate drugs for _SARS-CoV-2_. For all the COVID-19 nodes in our dataset
comprising 27 proteins (structured, nsp and orf), _SARS-CoV_ , _IBV_ , _MERS-
CoV_ , _CoV-229E_ , _CoV-NL63_ , and _MHV_ , we individually predict the drugs
for all these 33 entities. Each protein in _SARS-CoV-2_ targets a different
set of genes in humans, so we give individual predictions. We then pick the
top 10 drugs from all the predicted drugs and list 150 candidate repurposed
drugs for COVID-19. Out of these 150 drugs, 46 are currently in clinical
trials. Our predictions have a mixture of antivirals, antineoplastic,
corticosteroids, monoclonal antibodies (mAb), non-steroidal anti-inflammatory
drugs (NSAIDs), ACE inhibitors, and statin family of drugs, and some of the
vaccines discovered previously for other diseases. Refer to the Results
Section for a detailed discussion on the analysis of the predicted drugs for
COVID-19.

##  V Conclusions

In this work, we presented a generalized drug repurposing model, called Dr-
COVID for novel human diseases. We constructed a biological network of drugs,
diseases, genes, and anatomies and formulated the drug repurposing task as a
link prediction problem. We proposed a graph neural network model, which was
then trained to predict drugs for new diseases. Dr-COVID predicted 150
potential drugs for COVID-19, of which 46 drugs are currently in clinical
trials. The considered GNN model is computationally efficient and better ranks
known treatment drugs for diseases than the other GNN variants and non-deep
methods like the network proximity approaches. This work can be extended along
several directions. Considering the availability of substantial biological
data, the inclusion of information like individual side effects of drugs, the
molecular structure of the drugs, etc., may further improve the predictions.
Considering the comorbidities of a patient would help us analyze the
biological process and gene interactions in the body specific to an individual
and accordingly prescribe the line of treatment. Predicting a synergistic
combination of drugs for a disease would be another area of interest where
graph neural networks can be beneficial.

## Acknowledgements

S.P. Chepuri is supported in part by the Pratiskha Trust Young Investigator
Award, Indian Institute of Science, Bangalore, and the SERB grant
SRG/2019/000619, and S. Doshi is supported by the Robert Bosch Center for
Cyber Physical Systems, Indian Institute of Science, Bangalore, Student
Research Grant 2020-M-11 .

The authors thank the Deep Graph Learning team for compiling DRKG and making
the data public at https://github.com/gnn4dr/DRKG.

## Data and Code availability

All the implementation and the data required to reproduce the results in the
paper are available at https://github.com/siddhant-doshi/Dr-COVID.

## References

  * [1] World Bank. Global Economic Prospects, June 2020; 2020b.  http://hdl.handle.net/10986/33748.
  * [2] Zumla A, Chan JF, Azhar EI, Hui DS, Yuen KY.  Coronaviruses—drug discovery and therapeutic options.  Nat Rev Drug Discov. 2016;15(5):327–347.
  * [3] Paules CI, Marston HD, Fauci AS.  Coronavirus infections—more than just the common cold.  JAMA. 2020;323(8):707–708.
  * [4] Lu H, Stratton CW, Tang YW.  Outbreak of pneumonia of unknown etiology in Wuhan, China: The mystery and the miracle.  J Med Virol. 2020;92(4):401–402.
  * [5] Chen N, Zhou M, Dong X, Qu J, Gong F, Han Y, et al.  Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study.  Lancet. 2020;395(10223):507–513.
  * [6] Sohrabi C, Alsafi Z, O’Neill N, Khan M, Kerwan A, Al-Jabir A, et al.  World Health Organization declares global emergency: A review of the 2019 novel coronavirus (COVID-19).  Int J Surg. 2020;76:71–76.
  * [7] Pushpakom S, Iorio F, Eyers PA, Escott KJ, Hopper S, Wells A, et al.  Drug repurposing: progress, challenges and recommendations.  Nat Rev Drug Discov. 2018;18(1):41–58.
  * [8] Zitnik M, Nguyen F, Wang B, Leskovec J, Goldenberg A, Hoffman M.  Machine learning for integrating data in biology and medicine: Principles, practice, and opportunities.  Inf Fusion. 2019;50:71–91.
  * [9] Cheng F, Zhao Z.  Machine learning-based prediction of drug–drug interactions by integrating drug phenotypic, therapeutic, chemical, and genomic properties.  J Am Med Inform Assoc. 2014;21(e2):e278–e286.
  * [10] Yi Z, Li S, Yu J, Tan Y, Wu Q, Yuan H, et al.  Drug-drug interaction extraction via recurrent neural network with multiple attention layers.  International Conference on Advanced Data Mining and Applications. 2017;10604:554–566.
  * [11] Deng Y, Xu X, Qiu Y, Xia J, Zhang W, Liu S.  A multimodal deep learning framework for predicting drug-drug interaction events.  Bioinformatics. 2020;36(15):4316–4322.
  * [12] Zitnik M, Agrawal M, Leskovec J.  Modeling polypharmacy side effects with graph convolutional networks.  Bioinformatics. 2018;34(13):i457–i466.
  * [13] Cheng F, Desai RJ, Handy DE, Wang R, Schneeweiss S, Barábasi AL, et al.  Network-based approach to prediction and population-based validation of in silico drug repurposing.  Nat Commun. 2018;9(1):1–12.
  * [14] Guney E, Menche J, Vidal M, Barábasi AL.  Network-based in silico drug efficacy screening.  Nat Commun. 2016;7(1):1–13.
  * [15] Zhou Y, Hou Y, Shen J, Huang Y, Martin W, Cheng F.  Network-based drug repurposing for novel coronavirus 2019-nCoV/SARS-CoV-2.  Cell Discov. 2020;6(1):1–18.
  * [16] Cheng F, Lu W, Liu C, Fang J, Hou Y, Handy DE, et al.  A genome-wide positioning systems network algorithm for in silico drug repurposing.  Nat Commun. 2019;10(1):1–14.
  * [17] Gramatica R, Di Matteo T, Giorgetti S, Barbiani M, Bevec D, Aste T.  Graph theory enables drug repurposing–how a mathematical model can drive the discovery of hidden mechanisms of action.  PLOS One. 2014;9(1):e84912.
  * [18] Gysi DM, Valle ÍD, Zitnik M, Ameli A, Gan X, Varol O, et al.  Network medicine framework for identifying drug repurposing opportunities for COVID-19  arXiv:1403.3301 [Preprint]. 2020 [cited 2020 Nov 20]. Available from: https://arxiv.org/abs/2004.07229.
  * [19] Ioannidis VN, Zheng D, Karypis G.  Few-shot link prediction via graph neural networks for Covid-19 drug-repurposing  arXiv:2007.10261 [Preprint]. 2020 [cited 2020 Nov 20]. Available from: https://arxiv.org/abs/2007.10261.
  * [20] Rossi E, Frasca F, Chamberlain B, Eynard D, Bronstein M, Monti F.  SIGN: Scalable Inception Graph Neural Networks  arXiv:2004.11198 [Preprint]. 2020 [cited 2020 Nov 20]. Available from: https://arxiv.org/abs/2004.11198.
  * [21] Ioannidis VN, Song X, Manchanda S, Li M, Pan X, Zheng D, et al.  DRKG - Drug Repurposing Knowledge Graph for Covid-19  2020 [cited 2020 Nov 20]. Database: Github [Internet]. Available from: https://github.com/gnn4dr/DRKG/.
  * [22] Gordon DE, Jang GM, Bouhaddou M, Xu J, Obernier K, White KM, et al.  A SARS-CoV-2 protein interaction map reveals targets for drug repurposing.  Nature. 2020;583:459–468.
  * [23] Zaim S, Chong JH, Sankaranarayanan V, Harky A.  COVID-19 and multi-organ response.  Curr Prob Cardiology. 2020;45(8):100618.
  * [24] Zhang W, Zhao Y, Zhang F, Wang Q, Li T, Liu Z, et al.  The use of anti-inflammatory drugs in the treatment of people with severe coronavirus disease 2019 (COVID-19): The experience of clinical immunologists from China.  Clin Immunol. 2020;214:108393.
  * [25] Channappanavar R, Perlman S.  Pathogenic human coronavirus infections: causes and consequences of cytokine storm and immunopathology.  Semin Immunopathol. 2017;39(5):529–539.
  * [26] Chousterman BG, Swirski FK, Weber GF.  Cytokine storm and sepsis disease pathogenesis.  Semin Immunopathol. 2017;39(5):517–528.
  * [27] Jockusch S, Tao C, Li X, Anderson TK, Chien M, Kumar S, et al.  Library of Nucleotide Analogues Terminate RNA Synthesis Catalyzed by Polymerases of Coronaviruses Causing SARS and COVID-19.  Antiviral Res. 2020;180:104857.
  * [28] Caly L, Druce JD, Catton MG, Jans DA, Wagstaff KM.  The FDA-approved drug ivermectin inhibits the replication of SARS-CoV-2 in vitro.  Antiviral Res. 2020;178:104787.
  * [29] Fajgenbaum DC, Rader DJ.  Teaching old drugs new tricks: statins for COVID-19?  Cell Metab. 2020;32(2):145–147.
  * [30] Shin YH, Min JJ, Lee JH, Kim EH, Kim GE, Kim MH, et al.  The effect of fluvastatin on cardiac fibrosis and angiotensin-converting enzyme-2 expression in glucose-controlled diabetic rat hearts.  Heart Vessels. 2016;32(5):618–627.
  * [31] Zhang H, Penninger JM, Li Y, Zhong N, Slutsky AS.  Angiotensin-converting enzyme 2 (ACE2) as a SARS-CoV-2 receptor: molecular mechanisms and potential therapeutic target.  Intensive Care Med. 2020;46(4):586–590.
  * [32] Zhang XJ, Qin JJ, Cheng X, Shen L, Zhao YC, Yuan Y, et al.  In-hospital use of statins is associated with a reduced risk of mortality among individuals with COVID-19.  Cell Metab. 2020;32(2):176–187.
  * [33] Rega Institute, Belgian University KU Leuven.  KU Leuven Breakthrough as Modified Yellow Fever Virus Destroys COVID-19 in Preclinical Animal Research: Clinical Trials Next.  News article. 2020 [cited 2020 Nov 19]. Available from: https://www.trialsitenews.com/ku-leuven-breakthrough-as-modified-yellow-fever-virus-destroys-covid-19-in-preclinical-animal-research-clinical-trials-next/.
  * [34] Chen X, Chou CY, Chang GG.  Thiopurine analogue inhibitors of severe acute respiratory syndrome-coronavirus papain-like protease, a deubiquitinating and deISGylating enzyme.  Antivir Chem Chemother. 2009;19(4):151–156.
  * [35] Ragan I, Hartson L, Pidcoke H, Bowen R, Goodrich R.  Pathogen reduction of SARS-CoV-2 virus in plasma and whole blood using riboflavin and UV light.  PLOS One. 2020;15(5):e0233947.
  * [36] Weng JK.  Plant Solutions for the COVID-19 Pandemic and Beyond: Historical Reflections and Future Perspectives.  Mol Plant. 2020;13:803–807.
  * [37] Wishart DS, Feunang YD, Guo AC, Lo EJ, Marcu A, Grant JR, et al.  Drugbank 5.0: a major update to the Drugbank database for 2018.  Nucleic Acids Res. 2017;46(D1):D1074–D1082.
  * [38] Himmelstein DS, Lizee A, Hessler C, Brueggeman L, Chen SL, Hadley D, et al.  Systematic integration of biomedical knowledge prioritizes drugs for repurposing.  eLife. 2017;6:e26726.
  * [39] Percha B, Altman RB.  A global network of biomedical relationships derived from text.  Bioinformatics. 2018;34(15):2614–2624.
  * [40] Szklarczyk D, Gable AL, Lyon D, Junge A, Wyder S, Huerta-Cepas J, et al.  STRING v11: protein–protein association networks with increased coverage, supporting functional discovery in genome-wide experimental datasets.  Nucleic Acids Res. 2019;47(D1):D607–D613.
  * [41] Orchard S, Ammari M, Aranda B, Breuza L, Briganti L, Broackes-Carter F, et al.  The MIntAct project—IntAct as a common curation platform for 11 molecular interaction databases.  Nucleic Acids Res. 2014;42(D1):D358–D363.
  * [42] Cotto KC, Wagner AH, Feng YY, Kiwala S, Coffman AC, Spies G, et al.  GIdb 3.0: a redesign and expansion of the drug–gene interaction database.  Nucleic Acids Res. 2018;46(D1):D1068–D1073.
  * [43] Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P.  Geometric deep learning: going beyond euclidean data.  IEEE Signal Process Mag. 2017;34(4):18–42.
  * [44] Kipf TN, Welling M.  Semi-supervised classification with graph convolutional networks.  arXiv:1609.02907 [Preprint]. 2016 [cited 2020 Nov 20]. Available from: https://arxiv.org/abs/1609.02907.
  * [45] Hamilton W, Ying Z, Leskovec J.  Inductive representation learning on large graphs.  Advances in NeurIPS. 2017 (pp. 1024-1034).
  * [46] Veličković P, Cucurull G, Casanova A, Romero A, Lio P, Bengio Y.  Graph attention networks.  arXiv:1710.10903 [Preprint]. 2017 [cited 2020 Nov 20]. Available from: https://arxiv.org/abs/1710.10903.