11institutetext: Sergiu Ivanov 22institutetext: IBISC, Université Évry,
Université Paris-Saclay, France
22email<EMAIL_ADDRESS>33institutetext: Ion Petre
44institutetext: Department of Mathematics and Statistics, University of
Turku, Finland and 55institutetext: National Institute for Research and
Development in Biological Sciences, Romania
55email<EMAIL_ADDRESS>
# Controllability of reaction systems

Sergiu Ivanov  Ion Petre

(Received: date / Accepted: date)

###### Abstract

Controlling a dynamical system is the ability of changing its configuration
arbitrarily through a suitable choice of inputs. It is a very well studied
concept in control theory, with wide ranging applications in medicine,
biology, social sciences, engineering. We introduce in this article the
concept of controllability of reaction systems as the ability of transitioning
between any two states through a suitable choice of context sequences. We show
that the problem is -hard. We also introduce a model of oncogenic signalling
based on reaction systems and use it to illustrate the intricacies of the
controllability of reaction systems.

###### Keywords:

reaction systems controllability oncogenic signalling computational
complexity.

##  1 Introduction

Reaction systems are a biologically inspired model of computing originally
introduced in Ehrenfeucht:2007aa . They capture two fundamental interactions
typically present between biochemical entities—activation and inhibition.
Reaction systems are dynamical systems: reactions transform a set of reactants
into a set of products provided that none of its inhibitors are present, which
are then transformed further into other products, etc.

The reactions in reaction systems are governed by two fundamental principles:
the _threshold principle_ and the _non-permanency principle_. The threshold
principle stipulates that when a resource is available it is available in
unlimited amounts. This defines reaction systems as a qualitative modelling
framework, whose states are sets of species, without any quantitative
information. This also means that concurrency has to be modelled explicitly,
rather than implicitly via some intrinsic mechanisms of the modelling device.
The non-permanency principle states that if a resource is not explicitly
sustained/produced by reactions, it will disappear. The next state of a
reaction system only consists of the species explicitly produced by the
reactions enabled in the previous state.

Reaction systems are open systems: there is a notion of context that adds to
the current state in each step of its dynamic process. The next state is
produced by the reactions applied to the previous state plus the species added
by the context.

Since their introduction in 2007, two major research directions on reaction
systems have been established. The first direction focuses on their formal
properties as a dynamical systems: sequences of states
DBLP:journals/tcs/Salomaa12a , cSalomaa_Arto12a , power of small systems for
various size measures Formenti2015 , jSalomaa_Arto13b , cSalomaa_Arto14a ,
jSalomaa_Arto15c , TehAtanasiu2020 , cycles and attractors Azimi:2017aa ,
Dennunzio201996 , Formenti2014b , connections to propositional logic
DBLP:journals/ijfcs/Salomaa13 , etc. The second major direction of research on
reaction systems consists in exploring their potential as a modelling
framework, in particular for biological applications jAzIaPe14a , IF-RS ,
CorolliMMBM12 . This direction sparked interest in model checking for reaction
systems, i.e. formally defining relevant properties, evaluating their
complexity, and designing algorithms for checking them jAzGrIvMaPePo16a ,
jAzGrIvPe15a , Meski:2017aa , Meski2015 . These works revealed a whole wealth
of properties whose complexity ranges from polynomial to -hard.

In this paper, we continue the study of potential applications of reaction
systems to biological and medical research, and we focus on _controllability_.
Intuitively, controllability of a dynamical system is the ability of driving
this system to any one of its states, starting from any other state.
Controllability is a strong property, which has attracted a lot of attention,
especially in the case of biological networks (e.g., KolchHGK2015 , LiuSB2011
). Extensive research has been conducted into the practical feasibility of
different variants of controllability for biological networks, and exact and
approximate algorithms for finding ways to drive them, e.g. NetControlTCBB2018
, Gao:2014aa , Kanhaiya:2017aa , NetControlGeneticFrontiers . These results
have considerable potential for applications. For example, BadhwarBagler2015
identifies “driver neurons which can provide full control over the network”
governing the actions of the C. elegans worm. Also, art-lsb11 , Gao:2014aa ,
art-kcgp17 , MM3networks2020  discuss applications in drug repurposing and
personalised medicine.

The main goal of the present paper is defining controllability for reaction
systems, and establishing some computational complexity evaluations. Since
reaction systems are intrinsically open systems due to the context sequences
governing their evolution, introducing controllability is quite natural. In
addition to conventional, unrestricted controllability, we further define a
restricted variant, similar to the notion typically used in network control
theory, i.e., target controllability Gao:2014aa , NetControlTCBB2018 . We
illustrate our definitions on a novel, oncogenic signalling model that we
constructed based the Boolean model of G.-T.-Zanudo:2018aa . We show that
imposing restrictions on the context sequences or on the allowed observables
pushes the complexity of the controllability of reaction systems to -hard.

This paper is structured as follows. Section 2 recalls the basic notions, in
particular reaction systems, reachability, Boolean networks, and general
controllability. Section 3 introduces the running example we will use to
illustrate the notions of controllability, and which is translated from the
Boolean model in G.-T.-Zanudo:2018aa . Section 4 defines controllability for
reaction systems, and Section 5 defines target controllability. Finally,
Section 6 evaluates the complexity of the decision problems associated with
our definitions.

##  2 Preliminaries

###  2.1 Reaction systems

We recall in this section some of the basic concepts of reaction systems. The
presentation only aims to fix the notation and is kept brief. For details we
refer to Ehrenfeucht:2007aa , Ehrenfeucht:2016aa .

#### Basic definitions.

We introduce here the basic concepts around reaction systems.

###### Definition 1

Let $S$ be a finite (so-called background) set.

  * •

A _reaction_ in $S$ is a triplet $r=(R,I,P)$ such that $R,I,P\subseteq S$,
$R\cap I=\emptyset$. We call $R$, $I$, $P$ the _reactant_ , the _inhibitor_ ,
and the _product_ set of reaction $r$, resp. We also say that $R\cup I$ is the
_resource set_ of reaction $r$ and denote it $R\cup
I=\operatorname{\mathsf{rsc}}(r)$. For a set $A$ of reactions, its resource
set is $\operatorname{\mathsf{rsc}}(A)=\cup_{r\in
A}\operatorname{\mathsf{rsc}}(r)$.

  * •

Let $T\subseteq S$. We say that reaction $r=(R,I,P)$ is _enabled_ in $T$ if
$R\subseteq T$ and $I\cap T=\emptyset$. The _result_ of reaction $r$ on $T$ is
$\operatorname{\mathsf{res}}_{r}(T)=P$ if $r$ is enabled on $T$ and it is
$\operatorname{\mathsf{res}}_{r}(T)=\emptyset$ otherwise.

  * •

Let $R$ be a set of reactions in $S$ and $T\subseteq S$. The result of $R$ on
$T$ is $\operatorname{\mathsf{res}}_{R}(T)=\cup_{r\in
R}\operatorname{\mathsf{res}}_{r}(T)$.

  * •

A _reaction system_ $\mathcal{A}=(S,A)$ (over $S$) consists of a (finite) set
of reactions $A$ in $S$.

  * •

The _states_ of reaction system $\mathcal{A}=(S,A)$ are the subsets of $S$. We
say that state $V\subseteq S$ is _reachable_ in $\mathcal{A}$ if it is in the
domain of $\operatorname{\mathsf{res}}_{A}$, i.e.,
$V=\operatorname{\mathsf{res}}_{A}(U)$, for some $U\subseteq S$.

  * •

Let $\mathcal{A}=(S,A)$ be a reaction system and let $n\geq 1$. Let
$\gamma=C_{0},C_{1},\ldots C_{n}\subseteq S$ be a sequence of so-called
context sets. The _interactive process_ $\pi_{\gamma}(\mathcal{A})$ in
$\mathcal{A}$ defined by $\gamma$ consists of two state sequences of length
$n$
$\pi_{\gamma}(\mathcal{A})=(\delta_{\gamma}(\mathcal{A}),\tau_{\gamma}(\mathcal{A}))$
defined in the following way:

    * –

$\delta_{\gamma}(\mathcal{A})=D_{0},D_{1},\ldots,D_{n}$ is the _result
sequence_ of $\pi_{\gamma}(\mathcal{A})$ and
$\tau_{\gamma}(\mathcal{A})=W_{0},W_{1},\ldots,W_{n}$ is its _state sequence_
;

    * –

$D_{0}=\emptyset$, and
$D_{i}=\operatorname{\mathsf{res}}_{\mathcal{A}}(C_{i-1}\cup D_{i-1})$ for all
$i\in\\{1,\ldots,n\\}$;

    * –

$W_{i}=C_{i}\cup D_{i}$ for all $i\in\\{0,1,\ldots,n\\}$. $W_{0}$ is called
the _initial state_ of $\pi_{\gamma}(\mathcal{A})$. We say that
$\pi_{\gamma}(\mathcal{A})$ _starts_ in $W_{0}$ and _ends_ in $W_{n}$ and
denote it $W_{n}=\operatorname{\mathsf{res}}_{\mathcal{A}}^{\gamma}(W_{0})$
(the _result along context sequence $\gamma$_).

    * –

We say that state $V$ is _reachable_ from state $U$ if there is a context
sequence $\gamma$ such that
$V=\operatorname{\mathsf{res}}_{\mathcal{A}}^{\gamma}(U)$.

    * –

If $\gamma$ consists of empty (context) sets only, then
$\pi_{\gamma}(\mathcal{A})$ is called _context-independent_.

The dynamic processes defined by interactive processes can be seen as state
transition systems. We define the state transition graph of a reaction system
as the graph having as nodes the states of the reaction system and the edges
defined by the result function of the reaction system as follows.

###### Definition 2

Let $\mathcal{A}=(S,A)$ be a reaction system and $I\subseteq S$. The
$I$-context graph of reaction system $\mathcal{A}$ is the graph
$\mathcal{G}_{\mathcal{A}}^{I}=(\mathcal{P}(S),E)$, where the set of edges $E$
is $E=\\{(X,Y)\mid\exists C\subseteq
I:\operatorname{\mathsf{res}}_{\mathcal{A}}(X\cup C)=Y\\}$.

In the definition of the context graph we restrict the context sets to be
subsets of a given “input” set $I$ – this will be useful when we introduce the
notions of controllability for reaction systems. For $I=S$ there is no
constraint on the context sets and we obtain the usual transition system
associated to a reaction system, see Ehrenfeucht:2016aa .

Model checking for reaction systems has been considered in a number of
different setups, based on, e.g., temporal logic (Meski2015 , Meski:2017aa ,
Meski2019 ), and computational complexity (jAzGrIvPe15a , jAzGrIvMaPePo16a ,
Azimi:2017aa ). We recall here the result on the reachability problem
ReachabilityRS .

######  Theorem 2.1 (ReachabilityRS )

Deciding if a state $V$ of a reaction system $\mathcal{A}$ is reachable from a
state $U$ is -complete.

#### Boolean networks and reaction systems.

Multiple connections between Boolean functions and reaction systems have been
established, see BarbutiBGLM18 ; ATourOfRS ; Ehrenfeucht:2007aa . In what
concerns Boolean networks, different ways of defining the model exist, as for
example threshold Boolean networks (e.g., BarbutiBGLM18 ; VuongCIGT17 ), or
Boolean networks employing propositional formulae (e.g., BianeD19 ;
G.-T.-Zanudo:2018aa ). We only consider here the Boolean networks employing
propositional formulae, working in synchronous mode.

We recall a standard way of simulating a single Boolean function with reaction
systems, proposed in ATourOfRS ; Ehrenfeucht:2007aa . Let
$f:\mathbb{B}^{X}\to\mathbb{B}$ be a Boolean function, $f\notin X$, and
$\varphi$ a propositional formula in minimal disjunctive normal form
implementing $f$: $\varphi=\bigvee_{i}C_{i}$, where $C_{i}$ are the
conjunctions appearing in $\varphi$. We will use the notation $pos(C_{i})$ to
refer to the set of variables appearing without a negation in $C_{i}$ and
$neg(C_{i})$ to refer to the set of variables appearing with a negation in
$C_{i}$. Motivated by our biological running example, we also use a new
variable $\iota_{f}$ to $f$ that will be used to block the production of $f$
by inhibiting all reactions producing $f$. From the mathematical point of
view, these extra inhibitors are not needed for the equivalence between
Boolean networks and reaction systems.

The following set of reactions corresponds to $\varphi$:

$A_{\varphi}=\\{(pos(C_{i}),neg(C_{i})\cup\\{\iota_{f}\\},\\{f\\})\mid
C_{i}\in\varphi\\}.$

The reaction system
$\mathcal{A}_{\varphi}=(X\cup\\{f,\iota_{f}\\},A_{\varphi})$ simulates $f$ in
the following sense. Consider any truth assignment $s:X\to\mathbb{B}$ and
construct the corresponding subset of variables $W_{s}=\\{x\in X\mid
s(x)=1\\}$, using $s$ as a indicator function. Then
$\operatorname{\mathsf{res}}_{\mathcal{A}}(W_{s})=\\{f\\}$ if and only if
$f(s)=1$. Indeed, $\mathcal{A}_{\varphi}$ only produces $f$ on $W_{s}$ if and
only if there exists a conjunction $C_{i}\in\varphi$ for which
$pos(C_{i})\subseteq W_{s}$ and $neg(C_{i})\cap W_{s}=\emptyset$.

Consider now the Boolean network $BN=(f_{1},\dots,f_{n})$ over the set of
variables $X=\\{x_{1},\dots,x_{n}\\}$, in which $f_{i}$ is used to update the
variable $x_{i}$. The following set of reactions is obtained by iterating the
idea above for each Boolean function of the network:

$A_{BN}=\\{(pos(C),neg(C)\cup\\{\iota_{x_{i}}\\},\\{x_{i}\\})\mid
C\in\varphi_{i},1\leq i\leq n\\},$

where $\varphi_{i}$ denotes the Boolean formula in minimal disjunctive normal
form implementing the update function $f_{i}$. The reaction system
$\mathcal{A}_{BN}=(X,A_{BN})$ simulates the _synchronous_ model of dynamics of
the Boolean network $BN$.

###  2.2 Controllability of linear dynamical systems

We introduce here briefly the controllability of linear dynamical systems. For
a more detailed presentation we refer to NetControlTCBB2018 , Kanhaiya:2017aa
, NetControlGeneticFrontiers . We only discuss here a few basic concepts to
guide our definitions of the controllability of reaction systems. Intuitively,
a linear dynamical system consists of a set of nodes (variables) influencing
each other’s dynamics through linear, one-source/one-target interactions. They
can also be influenced through external, arbitrary interventions. The goal is
to be able to change the configuration of the system from any initial state to
any final state through a suitable choice of external interventions (that
depend on the initial and desired final state). A system having this property
is called controllable. A linear dynamical system is always trivially
controllable by adding external interventions on all its nodes, that
conveniently change its state as desired. The typical question to ask is one
of optimisation: given a linear dynamical system, find the minimal set of
external interventions making it controllable. We introduce these concepts
formally in the following.

A _linear dynamical system_ is a vector $x$ of functions
$x:\mathbb{R}\rightarrow\mathbb{R}^{n}$, $n\geq 1$, defined as the solution of
the system of ordinary differential equations

$\frac{dx(t)}{dt}=Ax(t),$

for some fixed initial value $x(0)\in\mathbb{R}^{n}$. The matrix $A$ defining
the dynamical system is an $n\times n$ real-valued matrix. The $(i,j)$ entry
of matrix $A$ describes the influence of the $j$-th node of the system over
its $i$-th node.

A linear dynamical system can also be influenced through an _external
contribution_ , thought of as a parametric $m$-dimensional vector $u$ of real
functions, influencing the $n$ nodes of the dynamical system through a matrix
$B\in\mathbb{R}^{n\times m}$. In this case, the linear dynamical system is
defined as the solution of the following system of ordinary differential
equations:

$\frac{dx(t)}{dt}=Ax(t)+Bu(t),$ (1)

and it is called the $(A,B)$ linear dynamical system.

One can also define a subset of so-called _target nodes_ of the dynamical
system on which the behaviour of the system is observed:
$T=\\{t_{1},t_{2},\ldots,t_{l}\\}\subseteq\\{1,2,\ldots,n\\}$, $1\leq l\leq
n$, $t_{i}<t_{j}$ for $1\leq i<j\leq n$. The set of target nodes can be
defined through its 0/1-valued characteristic matrix
$C_{T}\in\mathbb{R}^{l\times n}$ defined as follows: $C_{T}(i,j)=1$ if and
only if $t_{i}=j$. Obviously, if $T=\\{1,2,\ldots,n\\}$, then $C_{T}$ is the
identity matrix. A _targeted linear dynamical system_ is defined by a triplet
$(A,B,T)$.

We say that a dynamical system $(A,B)$ is _controllable_ if for any
$x(0)\in\mathbb{R}^{n}$ and any $\alpha\in\mathbb{R}^{n}$, there is an input
function vector $u_{x(0),\alpha}:\mathbb{R}\rightarrow\mathbb{R}^{n}$ such
that the solution $x$ of (1) satisfies the property $x(\tau)=\alpha$, for some
$\tau\geq 0$.

We say that a targeted dynamical system $(A,B,T)$ is _target controllable_ if
for any $x(0)\in\mathbb{R}^{n}$ and any $\gamma\in\mathbb{R}^{l}$, there is an
input function vector $v_{x(0),\alpha}:\mathbb{R}\rightarrow\mathbb{R}^{n}$
such that the solution $x$ of (1) satisfies the property
$C_{T}x(\tau)=\gamma$, for some $\tau\geq 0$. In other words, the solution
eventually matches $\gamma$ on its $T$-components. Obviously, for
$T=\\{1,2,\ldots,n\\}$, target controllability is identical to
controllability.

The (target) controllability problem has an elegant algebraic characterisation
known as Kalman’s condition.

######  Theorem 2.2 (Kalman’s condition Kalman1963 )

A targeted linear dynamical system $(A,B,T)$ is target controllable if and
only if its controllability matrix
$[C_{T}B,C_{T}AB,C_{T}A^{2}B,\ldots,C_{T}A^{n-1}B]$ is of full rank.

The controllability of linear dynamical system has found in the last few years
many applications in biology and medicine art-lsb11 , Gao:2014aa , Guo:2017aa
, art-kcgp17 , art-cgkkp18 . In this context the linear dynamical system is an
interaction (e.g., signalling) network describing the biological process of
interest, and the input is in terms of available drugs or small inhibitors.
The difficulty with this application of the concept is that the system is only
partially defined, with the majority of the interactions impossible to
measure, and thus with the matrices $A$ and $B$ only partially defined. The
solution is a structural formulation of the problem, where controllability is
defined in terms of the interaction network and not in terms of their precise
strength, see, e.g., art-cgkkp18 . Also, the problem in this context is often
given only through the interaction network (the equivalent of matrix $A$
above). The goal is to identify a suitable set of input nodes making the
system controllable, i.e., given matrix $A$, the problem is to identify a
suitable matrix $B$ such that the linear dynamical system $(A,B)$ is
controllable. Furthermore, for applications in medicine, with the input being
thought of as drugs delivered to a patient, there are various conditions
imposed on matrix $B$, such as having a minimal number of columns
(corresponding to minimising the number of drugs), or having the non-zero
entries of $B$ only on certain rows (corresponding to selecting the drugs only
from a certain set, e.g., FDA-approved drugs, or disease-specific standard
drugs). Some examples on applying the controllability problem in medicine are
in art-lsb11 , art-cgkkp18 , art-kcgp17 , MM3networks2020  . Software for
solving the controllability problem is available in netcontrol4biomed ,
NetControlGeneticFrontiers .

##  3 Running example: a reaction system model for breast cancer dynamics

Our running example is a reaction system modelling oncogenic signalling, with
a focus on the receptor tyrosine kinase (RTK) signalling network and on the
occurrence of uncontrolled proliferation. We follow the Boolean network model
proposed in G.-T.-Zanudo:2018aa  and give it a correspondent in terms of
reaction systems.

|
---|---
(a)

|  |
---|---|---
(b) | (c) | (d)

Figure 1: (a) The signal transduction network of G.-T.-Zanudo:2018aa
characterising cell proliferation. The $\operatorname{\mathsf{MAPK}}$ pathway
is in yellow, the $\operatorname{\mathsf{PIP3}}3$ pathway is in green, the
$\operatorname{\mathsf{AKT}}$-pathway is in dark green, the
$\operatorname{\mathsf{mTORC1}}1$-pathway is in orange, and the
$\operatorname{\mathsf{cycE/CDK2}}$-pathway is in blue. (b) - (d) Network
configurations leading to (b) uncontrolled proliferation, (c) proliferation,
and (d) no proliferation. In (b)-(d) a rectangle with blue background denotes
an inactive component, and one with yellow denotes an active component.

The model of G.-T.-Zanudo:2018aa  is grounded in the context of the breast
cancer model of Gomez-Tejeda-Zanudo:2017aa . It consists of a simplified
version of the RTK signalling network through the MAPK, PI3K, AKT, and mTORC1
pathways, including the cross-talk between them, and several feedback loops.
The network includes both oncogene proteins (RAS, PI3K, mTOR), as well as
tumour suppressors (Rb, FOXO3). The model is illustrated in Figure 1(a) (the
growth factors $\operatorname{\mathsf{GF}}$ are not included in the figure).
Each of the key proteins in Figure 1 has a correspondent variable in the
Boolean network model, with the update Boolean functions described in Table
LABEL:table-boolnet. Variable $X$ gets value $0$ if its correspondent protein
is inactive and $1$ if it is active. The outcome of the model is described by
variable $\operatorname{\mathsf{Prolif}}$ which, as an exception, is ternary-
valued, with $0$ standing for non-proliferation, $1$ for proliferation, and
$2$ for uncontrolled proliferation. Depending on the signals reaching the
output node, the model is in one of these three proliferation modes, as
described by the update function for variable
$\operatorname{\mathsf{Prolif}}$,
$f_{\operatorname{\mathsf{Prolif}}}=\operatorname{\mathsf{E2F}}2F+(\operatorname{\mathsf{EIF4F}}4F\land\operatorname{\mathsf{S6K}}6K)$.
Because of the non-binary nature of this function, we replace the modelling of
the proliferation status with two different variables,
$\operatorname{\mathsf{Prolif}}$ that models with true/false the
proliferation/non-proliferation status and $\operatorname{\mathsf{UProlif}}$
that models with true/false the uncontrolled proliferation status. Their
Boolean functions are defined as follows:

$\displaystyle f_{\operatorname{\mathsf{Prolif}}}$
$\displaystyle=(\operatorname{\mathsf{E2F}}2F\land\lnot\operatorname{\mathsf{EIF4F}}4F)\lor(\operatorname{\mathsf{E2F}}2F\land\lnot\operatorname{\mathsf{S6K}}6K)\lor(\lnot\operatorname{\mathsf{E2F}}2F\land\operatorname{\mathsf{EIF4F}}4F\land\operatorname{\mathsf{S6K}}6K),$
$\displaystyle f_{\operatorname{\mathsf{UProlif}}}$
$\displaystyle=\operatorname{\mathsf{E2F}}2F\land\operatorname{\mathsf{EIF4F}}4F\land\operatorname{\mathsf{S6K}}6K.$

Table 1: The Boolean network model of G.-T.-Zanudo:2018aa  for oncogenic
signalling in disjunctive normal form.
$f_{\operatorname{\mathsf{RTK}}}\,=\,(\operatorname{\mathsf{GF}}\land\operatorname{\mathsf{FOXO3}}3)\lor(\operatorname{\mathsf{GF}}\land\lnot\operatorname{\mathsf{S6K}}6K\land\lnot\operatorname{\mathsf{MAPK}})$
---
$f_{\operatorname{\mathsf{RAS}}}\,=\,\operatorname{\mathsf{RTK}}$
$f_{\operatorname{\mathsf{MAPK}}}\,=\,\operatorname{\mathsf{RAS}}\lor\operatorname{\mathsf{PIP3}}3$
$f_{\operatorname{\mathsf{PI3K}}3K}\,=\,\operatorname{\mathsf{RTK}}\lor\operatorname{\mathsf{RAS}}$
$f_{\operatorname{\mathsf{PIP3}}3}\,=\,\operatorname{\mathsf{PI3K}}3K$
$f_{\operatorname{\mathsf{FOXO3}}3}\,=\,\lnot\operatorname{\mathsf{AKT}}\lor\lnot\operatorname{\mathsf{MAPK}}$
$f_{\operatorname{\mathsf{AKT}}}\,=\,\operatorname{\mathsf{PIP3}}3$
$f_{\operatorname{\mathsf{cycE/CDK2}}}\,=\,(\operatorname{\mathsf{AKT}}\land\lnot\operatorname{\mathsf{FOXO3}}3)\lor\operatorname{\mathsf{E2F}}2F$
$f_{\operatorname{\mathsf{Rb}}}\,=\,\lnot\operatorname{\mathsf{cycE/CDK2}}$
$f_{\operatorname{\mathsf{E2F}}2F}\,=\,\lnot\operatorname{\mathsf{Rb}}$
$f_{\operatorname{\mathsf{TSC}}}\,=\,\lnot\operatorname{\mathsf{MAPK}}\lor\lnot\operatorname{\mathsf{AKT}}$
$f_{\operatorname{\mathsf{PRAS40}}40}\,=\,\lnot\operatorname{\mathsf{AKT}}$
$f_{\operatorname{\mathsf{mTORC1}}1}\,=\,\lnot\operatorname{\mathsf{TSC}}\land\lnot\operatorname{\mathsf{PRAS40}}40$
$f_{\operatorname{\mathsf{EIF4F}}4F}\,=\,{\operatorname{\mathsf{mTORC1}}1}$
$f_{\operatorname{\mathsf{S6K}}6K}\,=\,{\operatorname{\mathsf{mTORC1}}1}$
$f_{\operatorname{\mathsf{Prolif}}}\,=\,(\operatorname{\mathsf{E2F}}2F\land\lnot\operatorname{\mathsf{EIF4F}}4F)\lor(\operatorname{\mathsf{E2F}}2F\land\lnot\operatorname{\mathsf{S6K}}6K)\lor(\lnot\operatorname{\mathsf{E2F}}2F\land\operatorname{\mathsf{EIF4F}}4F\land\operatorname{\mathsf{S6K}}6K)$
$f_{\operatorname{\mathsf{UProlif}}}\,=\,\operatorname{\mathsf{E2F}}2F\land\operatorname{\mathsf{EIF4F}}4F\land\operatorname{\mathsf{S6K}}6K$

This model can reach a non-proliferation, a proliferation, and an uncontrolled
proliferation status, depending on the different activation statuses of the
MAPK-, PI3K-, AKT-, and mTORC1-pathways, see G.-T.-Zanudo:2018aa  and Figure
1(b)-(d) on page 1.

The reaction system associated to the Boolean network in Table LABEL:table-
boolnet has as its background set all the variables of the Boolean network and
an inhibitor variable associated to each of them:
$S=\\{\operatorname{\mathsf{GF}}$, $\operatorname{\mathsf{RTK}}$,
$\iota_{\operatorname{\mathsf{RTK}}}$, $\operatorname{\mathsf{RAS}}$,
$\iota_{\operatorname{\mathsf{RAS}}}$, $\operatorname{\mathsf{MAPK}}$,
$\iota_{\operatorname{\mathsf{MAPK}}}$, $\operatorname{\mathsf{PI3K}}3K$,
$\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\operatorname{\mathsf{PIP3}}3$,
$\iota_{\operatorname{\mathsf{PIP3}}3}$, $\operatorname{\mathsf{FOXO3}}3$,
$\iota_{\operatorname{\mathsf{FOXO3}}3}$, $\operatorname{\mathsf{AKT}}$,
$\iota_{\operatorname{\mathsf{AKT}}}$, $\operatorname{\mathsf{cycE/CDK2}}$,
$\iota_{\operatorname{\mathsf{cycE/CDK2}}}$, $\operatorname{\mathsf{Rb}}$,
$\iota_{\operatorname{\mathsf{Rb}}}$, $\operatorname{\mathsf{E2F}}2F$,
$\iota_{\operatorname{\mathsf{E2F}}2F}$, $\operatorname{\mathsf{TSC}}$,
$\iota_{\operatorname{\mathsf{TSC}}}$, $\operatorname{\mathsf{PRAS40}}40$,
$\iota_{\operatorname{\mathsf{PRAS40}}40}$, $\operatorname{\mathsf{mTORC1}}1$,
$\iota_{\operatorname{\mathsf{mTORC1}}1}$, $\operatorname{\mathsf{EIF4F}}4F$,
$\iota_{\operatorname{\mathsf{EIF4F}}4F}$, $\operatorname{\mathsf{S6K}}6K$,
$\iota_{\operatorname{\mathsf{S6K}}6K}$, $\operatorname{\mathsf{Prolif}}$,
$\iota_{\operatorname{\mathsf{Prolif}}}$, $\operatorname{\mathsf{UProlif}}$,
$\iota_{\operatorname{\mathsf{UProlif}}}\\}$. The set of reactions is listed
in Table LABEL:table-rsmodel.

In the reaction system model, the equivalent of a component/signal $X$,
$X\in\\{\operatorname{\mathsf{GF}}$, $\operatorname{\mathsf{RTK}}$,
$\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$,
$\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$,
$\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{AKT}}$,
$\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{Rb}}$,
$\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$,
$\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{mTORC1}}1$,
$\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K\\}$ being
active/inactive is having (not having, resp.) $X$ in the current state.
Synthetically activating/inactivating $X$ corresponds to the context adding to
the current state $X$ ($\iota_{X}$, resp.) A non-proliferation configuration
corresponds a state that does not include either symbol
$\operatorname{\mathsf{Prolif}}$ or $\operatorname{\mathsf{UProlif}}$. A
proliferation/uncontrolled proliferation configuration corresponds to a state
including $\operatorname{\mathsf{Prolif}}$ ($\operatorname{\mathsf{UProlif}}$,
resp.)

Table 2: A reaction systems model for oncogenic signalling.
$r_{\operatorname{\mathsf{RTK}}}^{(1)}\,=\,(\\{\operatorname{\mathsf{GF}},\operatorname{\mathsf{FOXO3}}3\\},\\{\iota_{\operatorname{\mathsf{RTK}}}\\},\\{\operatorname{\mathsf{RTK}}\\})$
---
$r_{\operatorname{\mathsf{RTK}}}^{(2)}\,=\,(\\{\operatorname{\mathsf{GF}}\\},\\{\operatorname{\mathsf{S6K}}6K,\operatorname{\mathsf{MAPK}},\iota_{\operatorname{\mathsf{RTK}}}\\},\\{\operatorname{\mathsf{RTK}}\\})$
$r_{\operatorname{\mathsf{RAS}}}\,=\,(\\{\operatorname{\mathsf{RTK}}\\},\\{\iota_{\operatorname{\mathsf{RAS}}}\\},\\{\operatorname{\mathsf{RAS}}\\})$
$r_{\operatorname{\mathsf{MAPK}}}^{(1)}\,=\,(\\{\operatorname{\mathsf{RAS}}\\},\\{\iota_{\operatorname{\mathsf{MAPK}}}\\},\\{\operatorname{\mathsf{MAPK}}\\})$
$r_{\operatorname{\mathsf{MAPK}}}^{(2)}\,=\,(\\{\operatorname{\mathsf{PIP3}}3\\},\\{\iota_{\operatorname{\mathsf{MAPK}}}\\},\\{\operatorname{\mathsf{MAPK}}\\})$
$r_{\operatorname{\mathsf{PI3K}}3K}^{(1)}\,=\,(\\{\operatorname{\mathsf{RTK}}\\},\\{\iota_{\operatorname{\mathsf{PI3K}}3K}\\},\\{\operatorname{\mathsf{PI3K}}3K\\})$
$r_{\operatorname{\mathsf{PI3K}}3K}^{(2)}\,=\,(\\{\operatorname{\mathsf{RAS}}\\},\\{\iota_{\operatorname{\mathsf{PI3K}}3K}\\},\\{\operatorname{\mathsf{PI3K}}3K\\})$
$r_{\operatorname{\mathsf{PIP3}}3}\,=\,(\\{\operatorname{\mathsf{PI3K}}3K\\},\\{\iota_{\operatorname{\mathsf{PIP3}}3}\\},\\{\operatorname{\mathsf{PIP3}}3\\})$
$r_{\operatorname{\mathsf{FOXO3}}3}^{(1)}\,=\,(\emptyset,\\{\operatorname{\mathsf{AKT}},\iota_{\operatorname{\mathsf{FOXO3}}3}\\},\\{\operatorname{\mathsf{FOXO3}}3\\})$
$r_{\operatorname{\mathsf{FOXO3}}3}^{(2)}\,=\,(\emptyset,\\{\operatorname{\mathsf{MAPK}},\iota_{\operatorname{\mathsf{FOXO3}}3}\\},\\{\operatorname{\mathsf{FOXO3}}3\\})$
$r_{\operatorname{\mathsf{AKT}}}\,=\,(\\{\operatorname{\mathsf{PIP3}}3\\},\\{\iota_{\operatorname{\mathsf{AKT}}}\\},\\{\operatorname{\mathsf{AKT}}\\})$
$r_{\operatorname{\mathsf{cycE/CDK2}}}^{(1)}\,=\,(\\{\operatorname{\mathsf{AKT}}\\},\\{\operatorname{\mathsf{FOXO3}}3,\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\},\\{\operatorname{\mathsf{cycE/CDK2}}\\})$
$r_{\operatorname{\mathsf{cycE/CDK2}}}^{(2)}\,=\,(\\{\operatorname{\mathsf{E2F}}2F\\},\\{\
\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\},\\{\operatorname{\mathsf{cycE/CDK2}}\\})$
$r_{\operatorname{\mathsf{Rb}}}\,=\,(\emptyset,\\{\operatorname{\mathsf{cycE/CDK2}},\iota_{\operatorname{\mathsf{Rb}}}\\},\\{\operatorname{\mathsf{Rb}}\\})$
$r_{\operatorname{\mathsf{E2F}}2F}\,=\,(\emptyset,\\{\operatorname{\mathsf{Rb}},\iota_{\operatorname{\mathsf{E2F}}2F}\\},\\{\operatorname{\mathsf{E2F}}2F\\})$
$r_{\operatorname{\mathsf{TSC}}}^{(1)}\,=\,(\emptyset,\\{\operatorname{\mathsf{MAPK}},\iota_{\operatorname{\mathsf{TSC}}}\\},\\{\operatorname{\mathsf{TSC}}\\})$
$r_{\operatorname{\mathsf{TSC}}}^{(2)}\,=\,(\emptyset,\\{\operatorname{\mathsf{AKT}},\iota_{\operatorname{\mathsf{TSC}}}\\},\\{\operatorname{\mathsf{TSC}}\\})$
$r_{\operatorname{\mathsf{PRAS40}}40}\,=\,(\emptyset,\\{\operatorname{\mathsf{AKT}},\iota_{\operatorname{\mathsf{PRAS40}}40}\\},\\{\operatorname{\mathsf{PRAS40}}40\\})$
$r_{\operatorname{\mathsf{mTORC1}}1}\,=\,(\emptyset,\\{\operatorname{\mathsf{TSC}},\operatorname{\mathsf{PRAS40}}40,\iota_{\operatorname{\mathsf{mTORC1}}1}\\},\\{\operatorname{\mathsf{mTORC1}}1\\})$
$r_{\operatorname{\mathsf{EIF4F}}4F}\,=\,(\\{\operatorname{\mathsf{mTORC1}}1\\},\\{\iota_{\operatorname{\mathsf{EIF4F}}4F}\\},\\{\operatorname{\mathsf{EIF4F}}4F\\})$
$r_{\operatorname{\mathsf{S6K}}6K}\,=\,(\\{\operatorname{\mathsf{mTORC1}}1\\},\\{\iota_{\operatorname{\mathsf{S6K}}6K}\\},\\{\operatorname{\mathsf{S6K}}6K\\})$
$r_{\operatorname{\mathsf{Prolif}}}^{(1)}\,=\,(\\{\operatorname{\mathsf{E2F}}2F\\},\\{\operatorname{\mathsf{EIF4F}}4F,\iota_{\operatorname{\mathsf{Prolif}}}\\},\\{\operatorname{\mathsf{Prolif}}\\})$
$r_{\operatorname{\mathsf{Prolif}}}^{(2)}\,=\,(\\{\operatorname{\mathsf{E2F}}2F\\},\\{\operatorname{\mathsf{S6K}}6K,\iota_{\operatorname{\mathsf{Prolif}}}\\},\\{\operatorname{\mathsf{Prolif}}\\})$
$r_{\operatorname{\mathsf{Prolif}}}^{(3)}\,=\,(\\{\operatorname{\mathsf{EIF4F}}4F,\operatorname{\mathsf{S6K}}6K\\},\\{\operatorname{\mathsf{E2F}}2F,\iota_{\operatorname{\mathsf{Prolif}}}\\},\\{\operatorname{\mathsf{Prolif}}\\})$
$r_{\operatorname{\mathsf{UProlif}}}\,=\,(\\{\operatorname{\mathsf{E2F}}2F,\operatorname{\mathsf{EIF4F}}4F,\operatorname{\mathsf{S6K}}6K\\},\\{\iota_{\operatorname{\mathsf{UProlif}}}\\},\\{\operatorname{\mathsf{UProlif}}\\})$

Table LABEL:t-RS-IP shows an interactive process of our reaction systems
model, running with the constant context $\\{\operatorname{\mathsf{GF}}\\}$,
and oscillating between proliferation and uncontrolled proliferation.

Table 3: An interactive process of the reaction systems model for oncogenic signalling.  Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
---|---|---|---
State |  $S_{1}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{mTORC1}}1\\}$ |  $S_{2}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{Prolif}}\\}$ |  $S_{3}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{Prolif}}\\}$
Status |  _No proliferation_ |  _Proliferation_ |  _Proliferation_
Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
State |  $S_{4}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40\\}$ |  $S_{5}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{Prolif}}\\}$ |  $S_{6}=\\{\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{mTORC1}}1\\}$
Status |  _No proliferation_ |  _Proliferation_ |  _No proliferation_
Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
State |  $S_{7}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K\\}$ |  $S_{8}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $S_{9}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$
Status |  _No proliferation_ |  _Uncontr. prolif._ |  _Uncontr. prolif._
Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
State |  $S_{10}=\\{\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $S_{11}=\\{\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $S_{12}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$
Status |  _Uncontr. prolif._ |  _Uncontr. prolif._ |  _Uncontr. prolif._
Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
State |  $S_{13}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{UProlif}}\\}$ |  $S_{14}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{Prolif}}\\}$ |  $S_{15}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{Prolif}}\\}$
Status |  _Uncontr. prolif._ |  _Proliferation_ |  _Proliferation_
Context |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$ |  $\operatorname{\mathsf{GF}}$
State |  $S_{16}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{Prolif}}\\}$ |  $S_{17}=\\{\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{Prolif}}\\}$ |  $S_{18}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PI3K}}3K$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{Prolif}}\\}$
Status |  _Proliferation_ |  _Proliferation_ |  _Proliferation_
Context |  $\operatorname{\mathsf{GF}}$ |  $\cdots$ |  $\cdots$
State |  $S_{19}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}=S_{8}$ |  aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aa $\dots$ |  aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aa $\dots$
Status |  _Uncontr. prolif._ |  $\cdots$ |  $\cdots$

##  4 Controllability of reaction systems

The basic concept of controllability of a reaction system $\mathcal{A}=(S,A)$
is that for any $X,Y\subseteq S$, $X\neq Y$, there is an interactive process
of $\mathcal{A}$ starting in $X$ and ultimately reaching $Y$. More exactly,
the controllability problem $(\mathcal{A},X,Y)$ is finding a context sequence
$\mathcal{C}=(C_{i})_{0\leq i\leq n}$ for $\mathcal{A}$ such that the
interactive process generated by $\mathcal{C}$ starts in $X$ and ends in $Y$.
$\mathcal{A}$ is said to be controllable if the controllability problem
$(\mathcal{A},X,Y)$ has a solution for any pair $X,Y\subseteq S$, $X\neq Y$.

Similarly as in the case of linear dynamical systems, in the absence of
constraints on the context sequences, the controllability problem has a
trivial solution: for any $X,Y\subseteq S$, the control sequence
$\mathcal{\gamma}_{X\to Y}=X,S,Y$ leads to an interactive process starting in
$X$, going to the empty state (since all reactions are disabled through
getting the full set $S$ as a context), and then going to $Y$:

Context |  $X$ |  $S$ |  $Y$
---|---|---|---
Result |  $\emptyset$ |  $\operatorname{\mathsf{res}}_{A}(X)$ |  $\emptyset=\operatorname{\mathsf{res}}_{A}(S)$
State |  $X$ |  $S$ |  $Y$

Instead, we define controllability of reaction systems as follows.

###### Definition 3

Let $\mathcal{A}=(S,A)$ be a reaction system.

  * •

For some nonnegative integer $n$ with $0\leq n<|S|$, we say that $\mathcal{A}$
is $n$-controllable if for any $X,Y\subseteq S$, with $Y$ reachable in
$\mathcal{A}$, there is a context sequence consisting of sets of cardinality
at most $n$ generating an interactive process starting in $X$ and ending in
$Y$.

  * •

For some $I\subseteq S$, we say that $\mathcal{A}$ is $I$-controllable if for
any $X,Y\subseteq S$, with $Y$ reachable in $\mathcal{A}$, there is a context
sequence consisting of subsets of $I$ generating an interactive process
starting in $X$ and ending in $Y$.

We define two versions of the controllability problem for reaction systems as
follows. Let $\mathcal{A}=(S,A)$ be a reaction system.

Problem $\mathcal{C}(\mathcal{A},n)$

For some nonnegative integer $n$ with $0\leq n<|S|$, decide if $\mathcal{A}$
is $n$-controllable. Also, find the smallest such $n$.

Problem $\mathcal{C}(\mathcal{A},I)$

For some $I\subseteq S$, decide if $\mathcal{A}$ is $I$-controllable. Also,
find a minimal (with respect to inclusion) such set $I$.

###### Example 1

The interactive process in Table LABEL:t-control-ex is an example on how to
induce a change of state from the attractor state $S_{18}$ (with uncontrolled
proliferation) in Table LABEL:t-RS-IP to state
$\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$,
$\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{FOXO3}}3$,
$\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$,
$\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$,
$\operatorname{\mathsf{Prolif}}\\}$ (with Proliferation). We use just one
additional symbol in the context sequence,
$\iota_{\operatorname{\mathsf{PI3K}}3K}$, inhibiting all reactions producing
$\operatorname{\mathsf{PI3K}}3K$.

Table 4: An interactive process of the reaction systems model for oncogenic signalling switching from an uncontrolled proliferation status to a proliferation status. Context |  $\\{\operatorname{\mathsf{GF}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$
---|---|---|---
State |  $X_{0}=S_{19}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $X_{1}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $X_{2}=\\{\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$
Status |  _Uncontr. prolif._ |  _Uncontr. prolif._ |  _Uncontr. prolif._
Context |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$
State |  $X_{3}=\\{\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $X_{4}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $X_{5}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{UProlif}}\\}$
Status |  _Uncontr. prolif._ |  _Uncontr. prolif._ |  _Uncontr. prolif._
Context |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}\\}$ |
State |  $X_{6}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{Prolif}}\\}$ |  $X_{7}=X_{6}$ |
Status |  _Proliferation_ |  _Proliferation_ |

##  5 Target controllability of reaction systems

The concept of target controllability of reaction systems is focused on a
given set of target nodes $T\subseteq S$ of a reaction system
$\mathcal{A}=(S,A)$. The objective in this problem is to be able, through a
suitable context sequence, to transition between any two subsets of $T$. The
caveat here is that, with the focus set on $T$ only, elements from $S\setminus
T$ may be present arbitrarily in the interactive process, including in the
initial and final states. We formalise this concept as follows.

###### Definition 4

Let $\mathcal{A}=(S,A)$ be a reaction system and $T\subseteq S$ the set of
targets.

  * •

For some nonnegative integer $n$ with $0\leq n<|S|$, we say that
$(\mathcal{A},T)$ is $n$-target controllable if for any $X,Y\subseteq T$, with
$Y$ or a superset of it reachable in $\mathcal{A}$, there is a context
sequence consisting of sets of cardinality at most $n$ generating an
interactive process starting in a state $W_{0}$ with $W_{0}\cap T=X$ and
ending in a state $W_{r}$ with $W_{r}\cap T=Y$, for some $r\geq 0$.

  * •

For some $I\subseteq S$, we say that $(\mathcal{A},T)$ is $I$-target
controllable if for any $X,Y\subseteq T$, with $Y$ or a superset of it
reachable in $\mathcal{A}$, there is a context sequence consisting of subsets
of $I$ generating an interactive process starting in a state $W_{0}$ with
$W_{0}\cap T=X$ and ending in a state $W_{r}$ with $W_{r}\cap T=Y$, for some
$r\geq 0$.

We define two versions of the target controllability problem for reaction
systems as follows. Let $\mathcal{A}=(S,A)$ be a reaction system and
$T\subseteq S$ the set of targets.

Problem $\mathcal{TC}(\mathcal{A},T,n)$

For some nonnegative integer $n$ with $0\leq n<|S|$, decide if
$(\mathcal{A},T)$ is $n$-target controllable. Also, find the smallest such
$n$.

Problem $\mathcal{TC}(\mathcal{A},T,I)$

For some $I\subseteq S$, decide if $(\mathcal{A},T)$ is $I$-target
controllable. Also, find a minimal (with respect to inclusion) such set $I$.

###### Example 2

A natural target of our running example is the proliferation symbol. Consider
for example driving the model away from the state $S_{18}$ in Table
LABEL:t-RS-IP, characterised by uncontrolled proliferation, and into a steady
state or an attractor consisting of states characterised by proliferation or
no proliferation. There are several ways of doing that with constant context
sequences that still include the $\operatorname{\mathsf{GF}}$ symbol,
including the following options:

  * •

the context set
$\\{\operatorname{\mathsf{GF}},\operatorname{\mathsf{PRAS40}}40\\}$ drives the
model into an attractor consisting of 10 states, all of them with
$\operatorname{\mathsf{Prolif}}$;

  * •

the context set
$\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$
drives the model into an attractor consisting of 11 states, 6 with
$\operatorname{\mathsf{Prolif}}$, and 5 without;

  * •

the context set
$\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{cycE/CDK2}}},\operatorname{\mathsf{PRAS40}}40\\}$
drives the model into an attractor consisting of 10 states, all of them with
no proliferation;

  * •

the context set
$\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K},\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$
drives the model into a steady state with no proliferation, with the
interactive process shown in Table LABEL:t-target-ex.

Table 5: An interactive process of the reaction systems model for oncogenic signalling switching from uncontrolled proliferation to a steady state with no proliferation. Context |  $\\{\operatorname{\mathsf{GF}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$
---|---|---|---
State |  $Y_{0}=S_{19}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{PIP3}}3$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{cycE/CDK2}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $Y_{1}=\\{\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{AKT}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $Y_{2}=\\{\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{E2F}}2F$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$
Status |  _Uncontr. prolif._ |  _Uncontr. prolif._ |  _Uncontr. prolif._
Context |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$
State |  $Y_{3}=\\{\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{mTORC1}}1$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{UProlif}}\\}$ |  $Y_{4}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{EIF4F}}4F$, $\operatorname{\mathsf{S6K}}6K$, $\operatorname{\mathsf{Prolif}}\\}$ |  $Y_{5}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40$, $\operatorname{\mathsf{Prolif}}\\}$
Status |  _Uncontr. prolif._ |  _Proliferation_ |  _Proliferation_
Context |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$ |  $\\{\operatorname{\mathsf{GF}},\iota_{\operatorname{\mathsf{PI3K}}3K}$, $\iota_{\operatorname{\mathsf{cycE/CDK2}}}\\}$ |
State |  $Y_{6}=\\{\operatorname{\mathsf{RTK}}$, $\operatorname{\mathsf{RAS}}$, $\operatorname{\mathsf{MAPK}}$, $\operatorname{\mathsf{FOXO3}}3$, $\operatorname{\mathsf{Rb}}$, $\operatorname{\mathsf{TSC}}$, $\operatorname{\mathsf{PRAS40}}40\\}$ |  $Y_{7}=Y_{6}$ |
Status |  _No proliferation_ |  _No proliferation_ |

##  6 Complexity results for the controllability of reaction systems

In this section we show that while unrestricted controllability is trivial, it
may become very complex if additional requirements are imposed on context
sequences. This observation is also valid for target controllability.

###  6.1 Controllability is -hard

We will show that $I$-controllability is at least as hard as reachability,
which is -complete ReachabilityRS .

###### Theorem 6.1

Let $\mathcal{A}=(S,A)$ be a reaction system and $I\subseteq S$. The problem
$\mathcal{C}(\mathcal{A},I)$ of deciding whether $\mathcal{A}$ is
$I$-controllable is -hard.

###### Proof

We will consider the particular case of $I$-controllability in which
$I=\emptyset\subseteq S$. Deciding that $\mathcal{A}$ is
$\emptyset$-controllable is equivalent to deciding whether, for any pair of
subsets $X,Y\subseteq S$, $Y$ is reachable from $X$. Since reachability for
reaction systems is -complete, we conclude that $\emptyset$-controllability
and therefore $I$-controllability of reaction systems is -hard.

Similarly, reachability can be reduced to $n$-controllability.

###### Theorem 6.2

Let $\mathcal{A}=(S,A)$ be a reaction system and $0\leq n<S$. The problem
$\mathcal{C}(A,n)$ of deciding whether $\mathcal{A}$ is $n$-controllable is
-hard.

###### Proof

As in the proof of Theorem 6.1, we consider the particular case $n=0$, which
only allows empty sets as contexts. In this case, $n$-controllability of
$\mathcal{A}$ is equivalent to deciding whether, for any pair of subsets
$X,Y\subseteq S$, $Y$ is reachable from $S$. This implies that
$n$-controllability is -hard.

###  6.2 Target controllability is -hard

Target controllability can be reduced to “full” controllability by allowing
any species to be a control target: $T=S$. This directly implies that
$I$-target controllability and $n$-target controllability are both -hard.

###### Corollary 1

Let $\mathcal{A}=(S,A)$ be a reaction system and $I,T\subseteq S$. The problem
$\mathcal{TC}(\mathcal{A},T,I)$ of deciding whether $\mathcal{A}$ is
$I$-controllable is -hard.

###### Corollary 2

Let $\mathcal{A}=(S,A)$ be a reaction system, $I\subseteq S$, and $0\leq
n\leq|S|$. The problem $\mathcal{TC}(\mathcal{A},T,n)$ of deciding whether
$\mathcal{A}$ is $n$-controllable is -hard.

These results are degenerate in the sense that they focus on the situations in
which control inputs are disallowed: $I=\emptyset$ or $n=0$. We will now show
that $I$-controllability for reaction systems is -hard even when some control
inputs _must_ be provided.

###### Theorem 6.3

Let $\mathcal{A}=(S,A)$ be a reaction system and $I,T\subseteq S$, such that
$I\neq\emptyset$. The problem $\mathcal{TC}(\mathcal{A},T,I)$ of deciding
whether $\mathcal{A}$ is $I$-controllable is -hard.

###### Proof

Consider an arbitrary reaction system $\mathcal{A}=(S,A)$, an extension of the
background set $S^{\prime}\supsetneq S$, and the reaction system
$\mathcal{A}^{\prime}=(S^{\prime},A)$ over the extended alphabet, but with the
same reactions as $\mathcal{A}$. Let $I=S^{\prime}\setminus S$ and $T=S$. Then
$(\mathcal{A}^{\prime},T)$ is $I$-target controllable if and only if
$\mathcal{A}$ is $\emptyset$-controllable.

Indeed, by construction the species from $I=S^{\prime}\setminus S$ have no
influence on the reactions in $A$. Therefore, given any pair of sets
$X,Y\subseteq S=T$, if $\mathcal{A}^{\prime}$ can reach $Y$ from $X$ with
contexts from $I$, it can reach $Y$ from $X$ with an empty context sequence.
Since $\mathcal{A}^{\prime}$ and $\mathcal{A}$ have exactly the same
reactions, this also means that if $\mathcal{A}^{\prime}$ can reach $Y$ from
$X$, $\mathcal{A}$ can reach $Y$ from $X$ as well. Conversely, if
$\mathcal{A}$ can reach $Y$ from $X$ with a sequence of empty contexts,
$\mathcal{A}^{\prime}$ can reach $Y$ from $X$ with a sequence of empty
contexts, or indeed with any sequence of contexts from $I$ of the same length.

We conclude that, for non-empty $I$, $I$-target controllability is at least as
hard as $I$-controllability, which proves the statement of the theorem.

The previous proof relies on restricting the set $I$ to symbols not actually
appearing in the reactions. We will now show that $I$-target controllability
is -hard even when contexts are allowed to inject symbols appearing in the
reactants and the inhibitors of reactions.

Before stating and proving this result, we formulate the following helper
notion. Consider the reaction system $\mathcal{A}=(S,A)$ and take an extension
of the background set $S^{\prime}\supseteq S$. We define the _nonce-extension_
$\operatorname{\mathsf{nonce}}_{S\to S^{\prime}}(A)$ of $A$ from $S$ to
$S^{\prime}$ in the following way:

$\operatorname{\mathsf{nonce}}_{S\to S^{\prime}}(A)=\\{(R\cup R^{\prime},I\cup
I^{\prime},P)\mid a\in A,R^{\prime}\subseteq S^{\prime}\setminus
S,I^{\prime}\subseteq S^{\prime}\setminus S\\}.$

Note that $A\subseteq\operatorname{\mathsf{nonce}}_{S\to S^{\prime}}(A)$,
because $\emptyset\subseteq S^{\prime}\setminus S$.

###### Theorem 6.4

Let $\mathcal{A}=(S,A)$ be a reaction system and $E,T\subseteq S$ such that
$E\cap\operatorname{\mathsf{rsc}}(A)\neq\emptyset$. The problem
$\mathcal{TC}(\mathcal{A},T,E)$ of deciding whether $\mathcal{A}$ is
$E$-controllable is -hard.

###### Proof

Consider a reaction system $\mathcal{A}=(S,A)$ and an extension of the
background set $S^{\prime}\supsetneq S$. Construct now a new reaction system
$\mathcal{A}^{\prime}=(S^{\prime},A^{\prime})$ with the properties $A\subseteq
A^{\prime}\subseteq\operatorname{\mathsf{nonce}}_{S\to S^{\prime}}(A)$. Let
$E=S^{\prime}\setminus S$ and $T=S$. Then $(\mathcal{A}^{\prime},T)$ is
$E$-target controllable if and only if $\mathcal{A}$ is
$\emptyset$-controllable. Stronger yet, for any $Z\subseteq S^{\prime}$,
$\operatorname{\mathsf{res}}_{\mathcal{A}^{\prime}}(Z)=\operatorname{\mathsf{res}}_{\mathcal{A}}(Z\cap
S)$. Indeed, suppose reaction $a^{\prime}=(R,I,P)\in A^{\prime}$ is enabled by
$Z$, i.e. $R\subseteq Z$ and $I\cap Z=\emptyset$. Then, trivially, the
reaction $a=(R\cap S,I\cap S,P)$ is enabled by $Z\cap S$, and $a\in A$ by
construction of $A^{\prime}$. On the other hand, if reaction $a\in A$ is
enabled by $Z\cap S$, then it is also enabled by $Z$, because $(Z\setminus
S)\cap(R\cup I)=\emptyset$.

##  7 Conclusions

The controllability problem is of high interest in dynamical systems, having
as its aim the ability to change the system’s configuration through well
chosen sequences of external interventions. We initiated in this paper the
study of controllability for reaction systems. The reaction systems framework
has all the key ingredients necessary for a natural definition of the
controllability problem: system dynamics through interactive processes, state
transitions, external interventions through context sequences. We defined
several natural variants of the controllability problem for reaction systems.

We introduced the first reaction system-based oncogenic signalling model in
the literature, a model that we believe will be of independent interest to the
reaction systems community. The model includes several of the best studied
cancer signalling pathways and follows their interplay leading to tumour
proliferation, both in the case of external growth factor signals, as well as
in their absence. We used this example to show how much diversity of options
there is in the concept of controllability for reaction systems. The
complexity of dynamics shown through this example anticipated the
computational complexity results we proved in this article, showing that the
controllability problem is -hard.

Several topics of interest remain to be explored around controllability of
reaction systems, for example, and in no particular order:

  1. 1.

Are there formulations of the problem that are computationally easy (in the
sense of computational complexity theory), perhaps based on minimal reaction
systems?

  2. 2.

Define and study the concept of stable controllability, where a constant (or
an ultimately constant) context sequence leads to the desired state, that
moreover is a steady state of the reaction system with the given constant
context sequence.

  3. 3.

Find efficient heuristics for the controllability of reaction systems,
identifying (not necessarily optimal) context sequences solving a given
controllability problem.

We believe that these topics should give further insight into the potential of
reaction systems as a qualitative framework for biomodelling.

## Funding

Sergiu Ivanov was partially supported by Computer Science Network of Paris
Île-de-France Region, project AAP DIM RFSI 2018-03. Ion Petre was partially
supported by the Romanian National Authority for Scientific Research and
Innovation (POC grant P_37_257 and PED grant 2391).

## Data availability

All data analysed in this study (the reaction systems model and the
interactive processes) is described in full in the article.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Sepinoud Azimi, Cristian Gratie, Sergiu Ivanov, Luca Manzoni, Ion Petre, and Antonio E. Porreca.  Complexity of model checking for reaction systems.  Theoretical Computer Science, 623:103–113, 2016.
  * [2] Sepinoud Azimi, Cristian Gratie, Sergiu Ivanov, and Ion Petre.  Dependency graphs and mass conservation in reaction systems.  Theoretical Computer Science, 598:23–39, 2015.
  * [3] Sepinoud Azimi, Bogdan Iancu, and Ion Petre.  Reaction system models for the heat shock response.  Fundamenta Informaticae, 131:1–14, 2014.
  * [4] Sepinoud Azimi, Charmi Panchal, Eugen Czeizler, and Ion Petre.  Reaction systems models for the self-assembly of intermediate filaments.  Annals of University of Bucharest, LXII(2):9–24, 2015.
  * [5] Sepinoud Azimi, Charmi Panchal, Andrzej Mizera, and Ion Petre.  Multi-stability, limit cycles, and period-doubling bifurcation with reaction systems.  International Journal of Foundations of Computer Science, 28(08):1007–1020, 2020/07/06 2017.
  * [6] Rahul Badhwar and Ganesh Bagler.  Control of neuronal network in caenorhabditis elegans.  PLoS One, 10,9 e0139204, 2015.
  * [7] Roberto Barbuti, Pasquale Bove, Roberta Gori, Francesca Levi, and Paolo Milazzo.  Simulating gene regulatory networks using reaction systems.  In Bernd-Holger Schlingloff and Samira Akili, editors, CS&P, volume 2240 of CEUR Workshop Proceedings. CEUR-WS.org, 2018.
  * [8] Célia Biane and Franck Delaplace.  Causal reasoning on boolean control networks based on abduction: Theory and application to cancer drug discovery.  IEEE/ACM Trans. Comput. Biology Bioinform., 16(5):1574–1585, 2019\.
  * [9] Robert Brijder, Andrzej Ehrenfeucht, Michael Main, and Grzegorz Rozenberg.  A tour of reaction systems.  International Journal of Foundations of Computer Science, 22(07):1499–1517, November 2011.
  * [10] Luca Corolli, Carlo Maj, Fabrizio Marini, Daniela Besozzi, and Giancarlo Mauri.  An excursion in reaction systems: From computer science to biology.  Theor. Comput. Sci., 454:95–108, 2012.
  * [11] Eugen Czeizler, Cristian Gratie, Wu Kai Chiu, Krishna Kanhaiya, and Ion Petre.  Structural target controllability of linear networks.  IEEE/ACM Transactions on Computational Biology and Bioinformatics, 15:1217–1228, 2018.
  * [12] Eugen Czeizler, Kai-Chiu Wu, Cristian Gratie, Krishna Kanhaiya, and Ion Petre.  Structural target controllability of linear networks.  IEEE/ACM Transactions on Computational Biology and Bioinformatics, 15(4):1217–1228, 2018.
  * [13] Alberto Dennunzio, Enrico Formenti, Luca Manzoni, and Antonio E. Porreca.  Reachability in resource-bounded reaction systems.  In AH. Dediu, J. Janous̆ek, C. Martín-Vide, and B. Truthe, editors, Language and Automata Theory and Applications. LATA 2016, volume 9618 of Lecture Notes in Computer Science. Springer, Cham, 2016.
  * [14] Alberto Dennunzio, Enrico Formenti, Luca Manzoni, and Antonio E. Porreca.  Complexity of the dynamics of reaction systems.  Information and Computation, 267:96 – 109, 2019.
  * [15] Andrzej Ehrenfeucht and Grzegorz Rozenberg.  Reaction systems.  Fundamenta Informaticae, 75(1):263–280, 2007.
  * [16] Andrzej Ehrenfeucht and Grzegorz Rozenberg.  Reaction Systems.  Fundamenta Informaticae, 75(1):263–280, January 2007.
  * [17] Enrico Formenti, Luca Manzoni, and Antonio E. Porreca.  Cycles and global attractors of reaction systems.  In Helmut Jürgensen, Juhani Karhumäki, and Alexander Okhotin, editors, Descriptional Complexity of Formal Systems, volume 8614 of Lecture Notes in Computer Science, pages 114–125. Springer International Publishing, 2014.
  * [18] Enrico Formenti, Luca Manzoni, and Antonio E. Porreca.  On the complexity of occurrence and convergence problems in reaction systems.  Natural Computing, 14(1):185–191, 2015.
  * [19] Jianxi Gao, Yang-Yu Liu, Raissa M. D’Souza, and Albert-László Barabási.  Target control of complex networks.  Nature Communications, 5(1):5415, 2014.
  * [20] Wei-Feng Guo, Shao-Wu Zhang, Ze-Gang Wei, Tao Zeng, Fei Liu, Jingsong Zhang, Fang-Xiang Wu, and Luonan Chen.  Constrained target controllability of complex networks.  Journal of Statistical Mechanics: Theory and Experiment, 2017(6):063402, 2017.
  * [21] R. E. Kalman, Y. C. Ho, and K. S. Narendra.  Controllability of linear dynamical systems.  Contributions to Differential Equations, 1:189–213, 1963.
  * [22] Krishna Kanhaiya, Eugen Czeizler, Cristian Gratie, and Ion Petre.  Controlling directed protein interaction networks in cancer.  Scientific Reports, 7(1):10327, 2017.
  * [23] Krishna Kanhaiya, Eugen Czeizler, Cristian Gratie, and Ion Petre.  Controlling directed protein interaction networks in cancer.  Scientific Reports, 7, 2017.
  * [24] Krishna Kanhaiya, Vladimir Rogojin, Keivan Kazemi, Eugen Czeizler, and Ion Petre.  Netcontrol4biomed: a pipeline for biomedical data acquisition and analysis of network controllability.  BMC Bioinformatics, 19, 2018.
  * [25] Walter Kolch, Melinda Halasz, Marina Granovskaya, and Boris N. Kholodenko.  The dynamic control of signal transduction networks in cancer cells.  Nature Reviews Cancer, 15:515–527, 2015.
  * [26] Yang-Yu Liu, Jean-Jacques Slotine, and Albert-László Barabási.  Controllability of complex networks.  Nature, 473:167–173, 2011.
  * [27] Yang-Yu Liu, Jean-Jacques Slotine, and Albert-László Barabási.  Controllability of complex networks.  Nature, 2011.
  * [28] Artur Mȩski, Maciej Koutny, and Wojciech Penczek.  Verification of linear-time temporal properties for reaction systems with discrete concentrations.  Fundamenta Informaticae, 154:289–306, 2017.
  * [29] Artur Mȩski, Maciej Koutny, and Wojciech Penczek.  Model checking for temporal-epistemic properties of distributed reaction systems.  Technical report, School of Computing, University of Newcastle upon Tyne, 2019.
  * [30] Artur Mȩski, Wojciech Penczek, and Grzegorz Rozenberg.  Model checking temporal properties of reaction systems.  Information Sciences, 2015.
  * [31] Victor-Bogdan Popescu, Krishna Kanhaiya, Iulian Năstac, Eugen Czeizler, and Ion Petre.  Identifying efficient controls of complex interaction networks using genetic algorithms.  Technical report, arXiv, arXiv:2007.04853 [q-bio.MN], 2020.
  * [32] Arto Salomaa.  Functions and sequences generated by reaction systems.  Theor. Comput. Sci., 466:87–96, 2012.
  * [33] Arto Salomaa.  On state sequences defined by reaction systems.  In Robert L. Constable and A. Silva, editors, Logic and Program Semantics, volume 7230 of Lecture Notes in Computer Science, pages 271–282. Springer-Verlag, 2012.
  * [34] Arto Salomaa.  Functional constructions between reaction systems and propositional logic.  Int. J. Found. Comput. Sci., 24(1):147–160, 2013.
  * [35] Arto Salomaa.  Minimal and almost minimal reaction systems.  Natural Computing, 12(3):369–376, 2013.
  * [36] Arto Salomaa.  Minimal reaction systems defining subset functions.  In Cristian Calude, Rusins Freivalds, and Kazuo Iwama, editors, Computing with New Resources, volume 8808 of Lecture Notes in Computer Science, pages 436–446. Springer, 2014.
  * [37] Arto Salomaa.  Two-step simulations of reaction systems by minimal ones.  Acta Cybernetica, 22(2):393–311, 2015.
  * [38] Jose Angel Sanchez Martin and Ion Petre.  Network controllability analysis of three multiple myeloma patient genetic mutation datatset.  Fundamenta Informaticae, 2020.
  * [39] Wen Chean Teh and Adrian Atanasiu.  Simulation of reaction systems by the strictly minimal ones, 2020.  To Appear in Journal of Membrane Computing.
  * [40] Quoc-Trung Vuong, Roselyne Chauvin, Sergiu Ivanov, Nicolas Glade, and Laurent Trilling.  A logical constraint-based approach to infer and explore diversity and composition in thresholded boolean automaton networks.  In Chantal Cherifi, Hocine Cherifi, Márton Karsai, and Mirco Musolesi, editors, Complex Networks & Their Applications VI - Proceedings of Complex Networks 2017 (The Sixth International Conference on Complex Networks and Their Applications), COMPLEX NETWORKS 2017, Lyon, France, November 29 - December 1, 2017, volume 689 of Studies in Computational Intelligence, pages 567–579. Springer, 2017.
  * [41] Jorge G. T. Zañudo, Maurizio Scaltriti, and Réka Albert.  A network modeling approach to elucidate drug resistance mechanisms and predict combinatorial drug treatments in breast cancer.  Cancer Converg, 1(1):5, 2017.
  * [42] Jorge G. T. Zañudo, Steven N. Steinway, and Réka Albert.  Discrete dynamic network modeling of oncogenic signaling: Mechanistic insights for personalized treatment of cancer.  Current Opinion in Systems Biology, 9:1–10, 2018.