2 Regulation of G1 / S Cell Cycle Transition
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2.5
2.0
TCP
1.5
TCP
GLY
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1.0
DCP
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DCP
0.0
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GDL
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2.0
TCP
TCP GLY
GLY
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1.0
0.5
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GDL
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80
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Time (s)
00
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70
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50
80
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Fig. 5. Numerical simulations for particular parameter values obtained as the outcome
of our framework. (Up left) Satisﬁable conﬁguration: DhaA = 0.0015, HheC = 0.007,
EchA = 0.01. (Up right) In-between conﬁguration: DhaA = 0.0035, HheC = 0.005,
EchA = 0.005. (Down left) Unsatisﬁable conﬁguration: DhaA = 0.01, HheC = 0.001,
EchA = 0.01. (Down right) Unsatisﬁable conﬁguration: DhaA = 0.01, HheC = 0.01,
EchA = 0.01. All values are in mM . Simulations were obtained in BIOCHAM [20].
EchA
Dh a A
Hh e C
Concentration (mM )
4.5
4.0
3.5
TCP
3.0
GLY
2.5
2.0
1.5
1.0
0.5
DCP
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60
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10
Time (s)
0
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0
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18
Fig. 6. (Left) Resulting parameter space for a speciﬁc initial state: TCP ∈
[3.84186, 5.0], DCP ∈ [0.0, 0.448898], GDL ∈ [0.0, 0.0669138], GLY ∈ [0.0, 0.01]. The
red dot shows the selected point for parameters values: DhaA = 0.001, HheC = 0.005,
EchA = 0.015. (Right) Numerical simulation for the selected point. All values are
in mM . Simulation was obtained in BIOCHAM [20]. (Color ﬁgure online)
High-Performance Symbolic Parameter Synthesis of Biological Models
93
and the central transcription factor E2F 1 (Fig. 7 (left)). For suitable parameter
values, two distinct stable attractors may exist (the so-called bistability). In [21]
a numerical bifurcation analysis of E2F 1 stable concentration depending on the
degradation parameter of pRB (φpRB ) has been provided. Note that traditional
methods for bifurcation analysis hardly scale to more than a single model parameter.
In this paper we demonstrate that by employing our algorithm we can provide bifurcation analysis for more than one parameter. In particular, we focus
on the synthesis of values of two interdependent parameters. We show how the
new results complement the results obtained with the algorithm employing the
interval-based representation of mutually independent parameters [9]. Additionally, we compare the results achieved within our workﬂow with the numerical
analysis provided in [21].
The property of bistability expresses that the system is able to settle in two
distinct stable states (i.e., levels of concentration) for speciﬁc initial conditions
and particular parameter values. It implies existence of a decision-making point
(or area) in the system.
The main outcome of the original analysis is shown in Fig. 8 (left) (produced by numerical analysis) displaying the dependency of stable concentration of E2F 1 on value of φpRB (degradation rate). The most interesting area
called unstable (for φpRB ∈ [0.007, 0.027]) determines feasible values of φpRB
wrt the above property. For φpRB < 0.007 the system converges to a lowerconcentration stable equilibrium whereas for φpRB > 0.027 it converges to a
higher-concentration stable equilibrium.
The CTL representation of the property in consideration is ϕ1 = (EF AG low
∧ EF AG high) where low = (0.5 < E2F 1 < 2.5) (representing safe cell behaviour) and high = (4 < E2F 1 < 7.5) (representing excessive cell division). During
the single run of our algorithm all subformulae of ϕ1 have been analysed. Let
ϕ2 = (AG low) and ϕ3 = (AG high) as the most interesting.
In [9] we have investigated perturbations of a single parameter φpRB with
the initial constraint φpRB ∈ [0.001, 0.025]. According to the Sect. 2 we have ﬁrst
created the PMA approximation of the original ODE model (Fig. 7 (right)) by
approximating each non-linear function in the right-hand side of ODEs with a
sum of optimal sequence of piecewise aﬃne ramp functions (the precision has
been set to 70 automatically generated segments per each non-linear function).
For such a setting the veriﬁcation process took less than 10 seconds on twenty
nodes. The results were processed by a Python script (Fig. 8 (right)). The plot
d[pRB]
dt
d[E2F 1]
dt
pRB
E2F1
J11
[E2F 1]
− φpRB [pRB]
m1 +[E2F 1] J11 +[pRB]
J12
a2 +[E2F 1]2
= k p + k2 2
− φE2F 1 [E2F 1]
Km2 +[E2F 1]2 J12 +[pRB]
= k1 K
a = 0.04, k1 = 1, k2 = 1.6, kp = 0.05, φpRB = 0.005
φE2F 1 = 0.1, J11 = 0.5, J12 = 5, Km1 = 0.5, Km2 = 4
Fig. 7. G1 /S transition regulatory network (left) and its ODE model (right).
E2F 1
M. Demko et al.
E2F 1
94
φpRB
φpRB
Fig. 8. (Left) Equilibrium curve for E2F 1 in proportion to φpRB as the result of
bifurcation analysis [21] (the authors conﬁrmed the scale of φpRB in the ﬁgure should
be 0.005-0.035 according to the text). (Right) Model checking results. Red and blue
are the high and low stable regions, respectively. Yellow are the states where ϕ1 holds.
(Color ﬁgure online)
intentionally depicts the same space as the Fig. 8 (left) to show obvious similarities of these results. The blue area stands for stable concentration of E2F 1 (yaxis) with particular value of φpRB (x-axis) satisfying the property ϕ2 , whereas
the red area satisﬁes the property ϕ3 . The yellow area (in the middle) stands
for possibility of reaching both stable concentrations. Due to mixing of existential and universal quantiﬁers (see Sect. 2), the results achieved for ϕ1 cannot be
exactly interpreted. On the contrary, the results for ϕ2 and ϕ3 are guaranteed
due to the conservativeness of the abstraction.
Although the algorithm based on interval-based encoding performs fast, it is
limited to independent parameters only. To overcome this limitation, we have
employed the SMT-based algorithm to explore two uncertain mutually dependent parameters. The method is computationally more demanding (about one
order of magnitude for each pair of dependent parameters). The goal of our
extended analysis is to explore the mutual eﬀect of the degradation parameter
of pRB (φpRB ) and the production parameter of pRB (k1 ) on the bistability.
Additionally, we perform post-processing of achieved results by employing additional constraints on the parameter space (e.g., imposing a lower and upper
bound on the production/degradation parameter ratio) and show an alternative
way of presenting the results.
In particular, we involve the SMT-based tool Symba [17] to obtain an approximated interval of the bounds on valid parameter values. Since the considered
parameters are linearly dependent, the resulting intervals cannot be simply combined to display the two-dimensional validity area in the parameter space. To this
end, we employ Symba to explore the ratio of the two parameters. By combining
initial parameter constraints with the bounds on the parameter ratio, a more
accurate parameter subspace is acquired. Such an outcome has been used with
the initial constraint φpRB ∈ [0.001, 0.1] and k1 ∈ [0.001, 10] (Fig. 9 (up left)).
High-Performance Symbolic Parameter Synthesis of Biological Models
k1
95
k1
φpRB
φpRB
k1
φpRB
φpRB
Fig. 9. (Up left) The resulting parameter space merged for all initial concentrations.
Each area corresponds to a diﬀerent property: ϕ1 (yellow), ϕ2 (blue) and ϕ3 (red). (Up
right) The same parameter space magniﬁed and projected to φpRB -axis. The framed
region agrees with the original numerical bifurcation analysis performed in [21] for
φpRB . (Down) Landscapes of the parameter space according to the quantitative satisfaction degree computed by BIOCHAM for ϕ2 (left) and ϕ3 (right), respectively. (Color
ﬁgure online)
Additionally, we have explored a reﬁned parameter space (φpRB ∈ [0.001, 0.025]
and k1 ∈ [0.001, 2]) where a one-dimensional projection on the φpRB -axis is
highlighted for k1 ≈ 1, the default value of k1 (Fig. 9 (up right)).
The analysis took 8 min on twenty nodes (excluding post-processing). The
obtained results can be used as a base for further analysis. We employ the feature
of BIOCHAM [10] to compute the landscape function that allows investigation of
quantitative satisfaction degree of the properties explored (Fig. 9 (down)). LTL
reformulation of ϕ2 and ϕ3 has been used (ϕ1 cannot be expressed in LTL). The
lighter is the colour the higher the satisfaction degree.
96
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M. Demko et al.
Conclusions
Recently developed methods for parameter synthesis of piecewise multi-aﬃne
systems have been embedded into a general workﬂow for biological models. The
workﬂow has been applied to a kinetic model of a synthetic metabolic pathway
and to a model of biological switch. In the former case, we have predicted admissible conﬁgurations of required enzymes concentration that guarantee the desired
production of glycerol under elimination of the toxicity. In the latter case, we
have obtained computationally eﬃcient analysis of bistability for two mutually
dependent parameters. In contrast to our previous results on synthesis of independent parameters, computational loads were signiﬁcantly increased. However,
the parallel algorithm was able to provide the results still in reasonable times
provided that an exhaustive amount of information about the systems dynamics
has been computed.
The main advantage is the global view of the systems dynamics. A disadvantage is the need for approximation and abstraction of the original ODE model.
For future work, it is important to integrate the results with the approximation
error and to make abstraction sensitive to the properties analysed.
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Influence Systems vs Reaction Systems
Fran¸cois Fages1(B) , Thierry Martinez2 , David A. Rosenblueth1,3 ,
and Sylvain Soliman1
1
3
Inria Saclay-ˆIle-de-France, Team Lifeware, Palaiseau, France
{Francois.Fages,Sylvain.Soliman}@inria.fr
2
Inria Paris, SED, Paris, France
Thierry.Martinez@inria.fr
Instituto de Investigaciones en Matem´
aticas Aplicadas y en Sistemas (IIMAS),
Universidad Nacional Aut´
onoma de M´exico (UNAM), Mexico, D.F., Mexico
drosenbl@unam.mx
Abstract. In Systems Biology, modelers develop more and more reaction-based models to describe the mechanistic biochemical reactions
underlying cell processes. They may also work, however, with a simpler formalism of inﬂuence graphs, to merely describe the positive and
negative inﬂuences between molecular species. The ﬁrst approach is promoted by reaction model exchange formats such as SBML, and tools
like CellDesigner, while the second is supported by other tools that have
been historically developed to reason about boolean gene regulatory networks. In practice, modelers often reason with both kinds of formalisms,
and may ﬁnd an inﬂuence model useful in the process of building a reaction model. In this paper, we introduce a formalism of inﬂuence systems
with forces, and put it in parallel with reaction systems with kinetics, in
order to develop a similar hierarchy of boolean, discrete, stochastic and
diﬀerential semantics. We show that the expressive power of inﬂuence
systems is the same as that of reaction systems under the diﬀerential
semantics, but weaker under the other interpretations, in the sense that
some discrete behaviours of reaction systems cannot be expressed by
inﬂuence systems. This approach leads us to consider a positive boolean
semantics which we compare to the asynchronous semantics of gene
regulatory networks `
a la Thomas. We study the monotonicity properties of the positive boolean semantics and derive from them an eﬃcient
algorithm to compute attractors.
1
Introduction
In Systems Biology, modelers develop more and more reaction models to describe
the biochemical reactions underlying cell processes. This approach is promoted
by reaction-model exchange formats such as SBML [18] and by the subsequent
creation of large reaction-based model repositories such as BioModels [25], without prejudging of their interpretation by diﬀerential equations, Markov chains,
Petri nets, or boolean transition systems [12].
Modelers can also work, however, with a simpler formalism of inﬂuence systems to merely describe the positive and negative inﬂuences between molecular
c Springer International Publishing AG 2016
E. Bartocci et al. (Eds.): CMSB 2016, LNBI 9859, pp. 98–115, 2016.
DOI: 10.1007/978-3-319-45177-0 7
Inﬂuence Systems vs Reaction Systems
99
species, without ﬁxing their implementation with biochemical reactions. In particular, boolean inﬂuence systems have been popularized in the 70’s by Glass,
Kauﬀman [15] and Thomas [30,31] to reason about gene regulatory networks,
represented by ordinary graphs between genes given with a boolean transition
table which deﬁnes their synchronous or asynchronous boolean transition semantics. Necessary conditions for multi-stability (cell diﬀerentiation) and oscillations
(homeostasis) have been given in terms of positive or negative circuits in the
inﬂuence graph [27,29]. Several tools such as GINsim [22], GNA [4] or Grifﬁn [28], use these properties and powerful graph-theoretic and model-checking
techniques to automate reasoning about the boolean state transition graph, compute attractors and verify various reachability and path properties. The representation of boolean inﬂuence systems by Petri nets was described in [6] but
leads to complicated encodings. It is also worth mentioning that inﬂuence systems with spatial information have been nicely developed in [7] as a formalism
particularly suitable for describing natural algorithms in life sciences and social
dynamics.
In Systems Biology, modelers often reason with both kinds of formalisms,
and may ﬁnd it useful to use and maintain an inﬂuence model in the process
of building a reaction model, for instance in order to reduce it while preserving
the essential inﬂuence circuits [23]. One reason is that it is easier to visualize
inﬂuence systems, rather than reaction systems for which complicated graphical
conventions such as SBGN [26] have been developed. While it is clear that the
inﬂuence graph is an abstraction of the reaction hypergraph [12], and perhaps
more surprisingly that the Jacobian inﬂuence system derived from the diﬀerential semantics of a reaction system is largely independent of the kinetics [13],
inﬂuence models are mostly used for their graphical representation and their
boolean semantics, but more rarely as a modeling paradigm for systems biology
with quantitative semantics using diﬀerential equations, or stochastic semantics.
In this paper, we introduce a formalism of inﬂuence systems with forces,
which we put in parallel with reaction systems with kinetics, in order to develop
a similar hierarchy of boolean, discrete, stochastic and diﬀerential semantics for
inﬂuence systems, similarly to what is done for reasoning about programs in the
framework of abstract interpretation [10,12]. We show that the expressive power
of inﬂuence systems is the same as that of reaction systems under the diﬀerential
semantics, but is weaker under the other interpretations, in the sense that some
formal discrete behaviours of reaction systems cannot be expressed by inﬂuence
systems. This approach provides an inﬂuence model with a hierarchy of possible
interpretations related by precise abstraction relationships, so that, for instance,
if a behavior is not possible in the boolean semantics, it is surely not possible in
the stochastic semantics whatever the inﬂuence forces are.
This leads us to consider a positive boolean semantics which we compare to
the asynchronous semantics of gene regulatory networks `
a la Thomas. We study
the monotonicity properties of the positive boolean semantics and derive from
them an eﬃcient algorithm to compute attractors. These concepts are illustrated
with models from the literature.
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F. Fages et al.
Preliminaries on Reaction Systems with Kinetics
In this article, unless explicitly noted, we will denote by capital letters (e.g. S)
sets or multisets, by bold letters (e.g., x ) vectors and by small roman or Greek
letters elements of those sets or vectors (e.g. real numbers, functions). For a multiset M , let Set(M ) denote the set obtained from the support of M , and brackets
like M (i) denote the multiplicity in the multiset (usually the stoichiometry). ≥
will denote the pointwise order for vectors, multisets and sets (i.e. inclusion).
2.1
Syntax
We recall here deﬁnitions from [11,13] for directed reactions with inhibitors:
Definition 1. A reaction over molecular species S = {x1 , . . . , xs } is a quadruple
(R, M, P, f ), also noted f for R/M ⇒ P , where R is a multiset of reactants, M
a set of inhibitors, P a multiset of products, all composed of elements of S, and
f : Rs → R, called kinetic expression, is a mathematical function over molecular
species concentrations. A reaction system is a ﬁnite set of reactions.
It is worth noting that a molecular species in a reaction can be both a reactant and a product (i.e. a catalyst), or both a reactant and an inhibitor (e.g.
Botts–Morales enzymes [19]). Such molecular species are not distinguished in
SBML and both are called reaction modiﬁers. Unlike SBML, we ﬁnd it useful to
consider only directed reactions (reversible reactions being represented here by
two reactions) and to enforce the following compatibility conditions between the
kinetic expression and the structure of a reaction.
Definition 2 ([11,13]). A reaction (R, M, P, f ) over molecular species
{x1 , . . . , xs } is well formed if the following conditions hold:
1. f (x1 , . . . , xs ) is a partially diﬀerentiable function, non-negative on Rs+ ;
2. xi ∈ R if and only if ∂f /∂xi (x) > 0 for some value x ∈ Rs+ ;
3. xi ∈ M if and only if ∂f /∂xi (x) < 0 for some value x ∈ Rs+ .
A reaction system is well formed if all its reactions are well formed.
Example 1. The classical prey-predator model of Lotka–Volterra can be represented by the following well-formed reaction system (without reaction inhibitors)
between a proliferating prey A and a predator B:
k1 * A * B for A + B = >2* B .
k2 * A for A = >2* A .
k3 * B for B = > _ .
Inﬂuence Systems vs Reaction Systems
2.2
101
Hierarchy of Semantics
As detailed in [12], a reaction system can be interpreted with diﬀerent formalisms that are formally related by abstraction relationships in the framework
of abstract interpretation [10] and form a hierarchy of semantics. We simply
recall here the deﬁnitions of the diﬀerent semantics of a reaction system.
The diﬀerential semantics corresponds to the association of an Ordinary
Diﬀerential Equation (ODE) system with the reactions in the usual way:
dxj
=
dt
(Pi (j) − Ri (j)) × fi
(Ri ,Mi ,Pi ,fi )
It is worth noting that in this interpretation, the inhibitors are supposed to
decrease the reaction rate but do not prevent the reaction from proceeding with
eﬀects on the products and reactants. For instance, in Example 2, we get the
classical Lotka–Volterra equations dB/dt = k1 ∗ A ∗ B − k3 ∗ B, dA/dt = k2 ∗
A − k1 ∗ A ∗ B, and the well-known oscillations between the concentrations of
the prey and the predator.
The stochastic semantics for reaction systems deﬁnes transitions between
discrete states describing numbers of each molecule, i.e. vectors x of Ns . A
transition is enabled if there are enough reactants, and the reaction propensity
is deﬁned by the kinetics:
∀(Ri , Mi , Pi , fi ), x −→fSi x with propensity fi if x ≥ Ri , x = x − Ri + Pi
Transition probabilities between discrete states are obtained through normalization of the propensities of all enabled reactions, and the time of next reaction
can be computed from the rates `
a la Gillespie [14]. In this interpretation, the
inhibitors are supposed to decrease the reaction propensity but do not prevent
the reaction from occurring. They are thus ignored here by the stochastic transition conditions as in the diﬀerential semantics. In Example 1, the stochastic
interpretation can exhibit some oscillations similar to the diﬀerential interpretation, and (almost surely) the extinction of the predator.
The discrete, or Petri Net, semantics is similar but ignores the kinetics and
is thus a trivial abstraction of the stochastic semantics by a forgetful functor:
∀(Ri , Mi , Pi , fi ), x −→D x if x ≥ Ri , x = x − Ri + Pi
The boolean semantics is similar to the discrete one but on boolean vectors x of Bs , obtained by the “zero, non-zero” abstraction of integers. With this
abstraction, when the number of a molecule is decremented, it can still remain
present, or become absent. It is thus necessary to take into account all the possible complete consumption or not of the reactants in order to obtain a correct
boolean abstraction of the discrete and stochastic semantics [12]. The boolean
transition system −→B is thus deﬁned by:
∀(Ri , Mi , Pi , fi ), ∀C ∈ P(Set(Ri )), x −→B x if x ⊇ Set(Ri ), x = x \C ∪Set(Pi )
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It is worth remarking that in Example 2 under this boolean interpretation,
one can observe either the stable coexistence of the prey and the predator, or
the extinction of the predator with or without the preceding extinction of the
prey.
As proven in [12], the last three of these semantics are related by successive
Galois connections, which means that if a behaviour is not possible in the boolean
semantics, it is not possible in the stochastic semantics whatever the reaction
kinetics are. On the other hand, the ﬁrst diﬀerential semantics is not an abstraction but rather a limit of the ﬁrst one for high number of molecules, as shown
for instance in [14].
It is worth noticing that the set of inhibitors of a reaction is just a syntactical
annotation which has not been used to deﬁne the diﬀerent semantics of the
hierarchy. One can also consider a boolean semantics with negation where the
set of inhibitors of a reaction is seen as a conjunction of negative conditions
for the transition (disjunctions can be represented with several reactions). The
boolean with negation transition system −→BN is then deﬁned by:
∀(Ri , Mi , Pi , fi )∀C ∈ P(Set(Ri ))x −→BN x
if x ⊇ Set(Ri ), x ∩ Mi = ∅, x = x \ C ∪ Set(Pi )
However, this strict interpretation of inhibitors by negations restricts the set of
possible boolean transitions and is not compatible with the diﬀerential semantics,
since in that interpretation an inhibitor may just slightly decrease the rate of a
reaction without preventing it from proceeding.
2.3
Influence Graph of a Reaction System
Here we recall two deﬁnitions of the inﬂuence graph associated with a reaction system, and their equivalence under general assumptions [11,13]. The ﬁrst
deﬁnition is based on the Jacobian matrix J formed of the partial derivatives
Jij = ∂ x˙i /∂xj , where x˙i is deﬁned by the diﬀerential semantics.
Definition 3. The diﬀerential inﬂuence graph associated with a reaction system is the graph having for vertices the molecular species, and for edge-set the
following two kinds of edges:
{A →+ B | ∂ x˙B /∂xA > 0 for some value x ∈ Rs+ }
∪{A →− B | ∂ x˙B /∂xA < 0 for some value x ∈ Rs+ }
Definition 4. The syntactical inﬂuence graph associated with a reaction system M is the graph having for vertices the molecular species, and for edges the
following set of positive and negative inﬂuences:
{A →+ B | ∃(Ri , Mi , Pi , fi ) ∈ M , (Ri (A) > 0 and Pi (B) − Ri (B) > 0)
or (A ∈ Mi and Pi (B) − Ri (B) < 0)}
∪{A →− B | ∃(Ri , Mi , Pi , fi ) ∈ M , (Ri (A) > 0 and Pi (B) − Ri (B) < 0)
or (A ∈ Mi and Pi (B) − Ri (B) > 0)}
The syntactical graph is trivial to compute, in linear time, by browsing the
syntax of the rules. Both deﬁnitions are equivalent under general assumptions: