=Paper=
{{Paper
|id=Vol-1373/paper6
|storemode=property
|title=Dependent shrink for Petri net models of signaling pathways
|pdfUrl=https://ceur-ws.org/Vol-1373/paper6.pdf
|volume=Vol-1373
|dblpUrl=https://dblp.org/rec/conf/apn/MizutaGM15
}}
==Dependent shrink for Petri net models of signaling pathways==
https://ceur-ws.org/Vol-1373/paper6.pdf
Dependent shrink for Petri net models of
signaling pathways
Atsushi Mizuta1 , Qi-Wei Ge2 , and Hiroshi Matsuno1
1
Graduate School of Science and Engineering, Yamaguchi University
Yamaguchi 753-8512, Japan
2
Faculty of Education, Yamaguchi University
Yamaguchi 753-8512, Japan
{v014vc,gqw}@yamaguchi-u.ac.jp, matsuno@sci.yamaguchi-u.ac.jp
Abstract. Retention-free Petri net has been used in modeling of sig-
naling pathways, which is a timed Petri net such that total input and
total output token flows are equivalent at any place. Previously we have
investigated the dependency of transitions in retention-free Petri net. In
this paper, we introduce a modeling method for signaling pathway by
using Petri net, giving properties of retention-freeness by considering arc
weight. Based on the obtained properties, we propose an algorithm to
find shrinkable transitions and to shrink them into a single transition.
This algorithm eventually provides a set of transitions whose firing fre-
quencies are dependent. As an example, we apply the algorithm to IL-3
signaling pathway Petri net model to show the usefulness of our proposed
algorithm.
Keywords: signaling pathway, Petri net, retention-free Petri net, de-
pendent shrink
1 Introduction
Li et al. [1] have proposed a qualitative modeling method by paying attention
to the molecular interactions and mechanisms using discrete Petri nets. Further-
more, Miwa et al. [2] modeled it with timed Petri net, which is an extended
Petri net on the concept of time, proposing a method to have firing frequency
conditional expressions based on its structure information. At the same time,
they introduced “retention-free” Petri net for defining smooth signal flows in
signaling pathways.
In Petri net model of signaling pathway, firing frequency of each transition
should be measured by biological experiments. However, such biological data
of reactions are very few. As a method to cope with this problem, Murakami
et al. [3] proposed an approach to check the retention-freeness of a given Petri
net based on firing frequencies of transitions of this Petri net. According to
this method, Matsumoto et al. [4] formally described the concept of dependent
shrink after giving formal definitions of dependent subnet. Dependent shrink is
a concept to express a dependent subnet which is shrunk into a single transition.
M Heiner, AK Wagler (Eds.): BioPPN 2015, a satellite event of PETRI NETS 2015,
CEUR Workshop Proceedings Vol. 1373, 2015.
86 A Mizuta, QW Ge and H Matsuno
Place Transition Directed arc
Fig. 1. Elements of Petri nets
The advantage of this concept is that all the firing frequencies of transitions in
the subnet can be computationally obtained from the firing frequency of that
shrunk transition. Namely, by only getting the reaction speed of a reaction that
corresponds to that shrunk transition, all other reaction speeds in dependent
subnet can be estimated by the proposed procedure in this paper.
In this paper we propose an algorithm to do equivalent transformation for
retention-free Petri nets. Concretely, we first classify dependent shrink pattern
according to the patterns of input and output transitions of a place and then
perform the dependent shrink operations of the patterns. Finally, we reconstruct
the Petri net based on the dependent shrink result.
2 Basic Definitions and Properties
In this section, we briefly give the necessary definitions and properties of Petri
nets. For detailed definitions the reader is suggested to refer to [5].
Definition 1. A Petri net denoted as P N = (T, P, E, α, β) that is a bipartite
graph, where E = E + ∪ E − and
– T : a set of transitions {t1 , t2 , · · · , t|T | }
– P : a set of places {p1 , p2 , · · · , p|P | }
– E + : a set of arcs from transitions to places e = (t, p)
– E − : a set of arcs from places to transitions e = (p, t)
– αe : is the weight of arc e = (p, t)
– βe : is the weight of arc e = (t, p)
Definition 2. Let P N be a Petri net
1. • t (t• ) is the set of input (or output) places of t, and • p (p• ) is the set of
input (or output) transitions of p.
2. A transition without input arc is called source transition and the set of
source transitions are denoted by Tsour = {tsour 1 , · · · , tsour
a }(a ≥ 1).
3. A transition without output arc is called sink transition and the set of sink
transitions is denoted by Tsink = {tsink 1 , · · · , tsink
b }(b ≥ 1).
4. A transition t is called Ps − synchronous transition if there exists a set
of input places Ps that for any p ∈ Ps , p• = {t} holds, and is defined by
Tsync = {tsync
1 , · · · , tsync
c }(c ≥ 1).
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Dependent shrink for signaling pathways 87
5. A place can hold a positive integer that represents a number of tokens. An
assignment of tokens in places expressed in form of a vector M is called a
marking, which varies during the execution of a Petri net. Given with an
initial marking M0 , the Petri net is called Marked Petri net and denoted by
M P N = (P N, Mo ).
Fig. 2 shows source(i) and sink(ii) transition and Fig. 3 shows synchronous
transitions. Note that, we use discrete Petri nets in this paper.
Fig. 2. Source and sink transition
p1 p3
t1
p2 p4
Fig. 3. Synchronous transition
2.1 Modeling rules
Li et al. [1] gave the following modeling rules for signaling pathways based on
Petri net representation.
1. Places denote static elements including chemical compounds, conditions,
states, substances, and cellular organelles participating in the biological
pathways. Tokens indicate the presence of these elements. The number of
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88 A Mizuta, QW Ge and H Matsuno
tokens can be regarded as a representation of the amount of chemical sub-
stances. Current assignment of tokens to the places is expressed in form of
a vector, namely a marking as defined above.
2. Transitions denote active elements including chemical reactions, events, ac-
tions, conversions, and catalyzed reactions. A transition fires by taking off
tokens from its individual input places and creating new tokens that are dis-
tributed to its output places if its input places has at least as many tokens
in it as arc weight from the place to the transition.
3. Directed arcs connecting the places and the transitions represent the rela-
tions between corresponding static elements and active elements. Arc weights
α and β (defined in Definition 1) describe the quantities of substances re-
quired before and after a reaction, respectively. Especially in case of modeling
a chemical reaction, arc weights represent quantities given by stoichiometric
equations of the reaction itself. Note that, weight of an arc is omitted if the
weight is 1.
4. Since an enzyme itself plays a role of catalyzer in biological pathways and
there occurs no consumption in biochemical reactions, an enzyme is excep-
tionally modeled in Definition 3 below.
5. An inhibition function in biological pathways is modeled by an inhibitor arc.
Definition 3. [1] An enzyme in a biological pathway is modeled by a place,
called enzyme place, as shown in Fig. 4.
1. Enzyme place pi has a self-loop with the same weight connected from and to
transition ts . Once an enzyme place is occupied by a token, the token will
return to the place again to keep the firable state, if the transition ts is fired.
2. Let tp and td denote a token provider of pi and a sink output transition
of pi , respectively, where the firing of tp represents an enzyme activation
reaction and the firing of td implies a small natural degradation in a biological
pathway. pi holds up token(s) after firing transition tp and the weights of the
arcs satisfy α(pi , td ) α(pi , ts ).
Fig. 4. An enzyme place in Petri net model
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Dependent shrink for signaling pathways 89
[Firing rule of Petri net] A transition t is firable if each input place pI of P N
has at least αe tokens, where αe denotes the weight of an arc e = (pI , t). Firing
of a transition t removes αe tokens from each input place pI of t and deposit
βe (e = (t, pO )) tokens to each output place pO of t, where βe denotes the weight
of an arc e = (t, pO ). A source transition is always firable.
Definition 4. A timed Petri net T P N is defined by T P N = (P N, D), where D
is a set of positive number expressing firing delay times (or delay time for short)
of transitions in T .
[Firing rule of timed Petri nets] (i) If the firing of a transition ti is decided,
tokens required for the firing are reserved. We call these tokens as reserved
tokens. (ii) When the delay time di of ti passed, ti fires to remove the reserved
tokens from the input places of ti and put non-reserved tokens into the output
places of ti . In a timed Petri net, firing times of a transition ti per unit time is
called f iring f requency fi . fi represents the maximum firing frequency of ti .
The delay time di of ti is given by the reciprocal of fi .
Definition 5. [2] With the firing of transition tI , token amounts flowed into
place p per unit time is called “input token-flow”, and is denoted by T FtI ,p . On
the other hand, with the firing of transition tO , token amounts flowed out of place
p per unit time is called “output token-flow”, and is denoted by T Fp,tO . T FtI ,p
and T Fp,tO (shown in Fig. 5) are defined by following equations, respectively:
T FtI ,p = fI · βI (1)
T Fp,tO = fO · αO , (2)
where fI and fO are firing frequencies of tI and tO , respectively; βI and αO are
the weights of e = (tI , p) and e = (p, tO ), respectively.
Based on this definition, the following equation hold.
•
Proposition 1. [2] Let p be a place P
•
Pn {tIi |tIi ∈ p} and
with input transitions
m
out put transitions {tOj |tOj ∈p }. Then i=1 T FtIi ,p and j=1 T Fp,tOj are the
total input token-flow and the total output token-flow for place p, respectively.
Furthermore, when firing frequency f take the maximum firing rate f , input
token-flow T FtI ,p and output token flow T Fp,tO become the maximum, F TtI ,p and
F Tp,tO , respectively. These maximum token-flows satisfy following equations.
m
X m
X
T FtIi ,p ≤ F TtIi ,p (3)
i=1 i=1
n
X Xn
T Fp,tOj ≤ F Tp,tOj (4)
j=1 j=1
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90 A Mizuta, QW Ge and H Matsuno
Fig. 5. Token flows
The following requirement is trivial.
Proposition 2. [2] In a timed Petri net, a total output token-flow is not more
than a total input token-flow for each place p:
m
X n
X
T FtIi ,p ≥ T Fp,tOj , (5)
i=1 j=1
Definition 6. [2] A timed Petri net TPN is called Retention-free Petri net
(RFPN) (satisfying Proposition1) if a total input token-flow and a total output
token-flow are equivalent at any place of TPN; that is,
T FtIi ,p = T Fp,tOj (6)
Definition 7. [3] Each unreserved token deposited to input place p is assigned
to be reserved by the transition tOj that satisfies the following expression:
n
cj X ck
{ / − sj } =
αj αk
k=1
n
ci X ck
min{ / − si | i = 1, 2, · · · , n} (7)
αi αk
k=1
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Dependent shrink for signaling pathways 91
When the number of reserved tokens of tOj is not less than a required token
number for the firing, the firing of tOj is decided. After the delay time dOj of
tOj passed, tOj fires to remove the reserved tokens from the input place of tOj
and deposit unreserved tokens into the output places of tOj .
In the above expression (7), sj is the firing probability of transition tOj , which
represents the proportion of the firing frequency of each transition to the total
firing frequencies of the transitions in conflict. A probability sj is assigned to
corresponding transition tOj , which is given as a constant in advance according
to the event. A variable c is an accumulated number of tokens that tOj has been
c
reserved so far, and thus b αjj c represents the number of firing times of transition
tOj from the beginning.
Expression (7) is designed to reserve the token to such a transition Pn ti that
c
has the largest difference between calculated firing probability αjj / k=1 αckk and
given firing probability sj among all the transitions in conflict.
Definition 8. [2] If output transitions of p are in conflict, the maximum firing
frequency of tOj must satisfy the following expression:
m
s ·α
Pn j j
X
· T FtIi ,p = fOj · αj , (8)
k=1 sk · αk i=1
where αj is the weight of e = (p, tOj ) and sj is the firing probability of tOj .
Pnsj ·αj represents the ratio of the token amount deposited to tOj to the total
k=1 sk ·αk
token-flow from p to each output transition p• .
3 Shrink of Dependent Subnet
The equation (8) shows a relationship of firing frequency about input and output
transitions, which are dependent each other. Based on this dependency, a set of
transitions can be obtained, by which firing frequencies of all transitions in a
Petri net model can be calculated. Note that the transitions in the set determined
in this way correspond to the reactions whose speeds need to be measured by
biological experiments.
3.1 Dependent subnet
Dependent subnet, obtained as follows, is a Petri net induced from a set of
transitions which are dependent on each other.
Definition 9. [4] If firing frequency of a transition t is determined by the firing
frequency of transition α, this transition is called α-dependent transition. The
subnet induced by the set of α-dependent transitions and transition α is called
α-dependent subnet, denoted by P Nα .
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92 A Mizuta, QW Ge and H Matsuno
Definition 10. [4] For a given set A of transitions, A-dependent transition is
a set of transitions whose firing frequencies are determined by the firing frequen-
cies of transitions in A. The subnet induced by A-dependent transition and A is
called A-dependent subnet, denoted by P NA .
3.2 Dependent shrink
For a set of α-dependent transition T , dependent shrink is a procedure to sub-
stitute the set of α-dependent transition T to a single transition t.
Definition 11. [4] If two transitions ti and tj exist are dependent each other,
these two transitions can be shrunk into a single transition.
Note that the proofs of the following propositions are omitted to save the space
of this paper.
Proposition 3. As shown in Fig. 6, if a place p has one input transition tI
and one output transition tO , these two transitions can be shrunk into a single
transition t0 , where the weight of new input arc α0 = α1 , and the weight of new
output arc β 0 = β2 · αβ12 .
Fig. 6. Markgraph
Proposition 4. As shown in Fig. 7, if place p has multiple output transitions
TO = {tO.1 , tO.2 , · · · , tO.k }, TO can be shrink into a single transition t0 , where
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Dependent shrink for signaling pathways 93
the weights of input arc α0 and new output arc β 0 are defined by the following
formulas;
sO.1 · α1 + sO.2 · α2 + · · · + sO.k · αk
α0 = (9)
sO.1
β10 = sO.1 · β1
β20 = sO.2 · β2
..
.
0
βk = sO.k · βk (10)
Fig. 7. Conflict structure
Proposition 5. As shown is Fig. 8, if place p has multiple output transitions
of two types, with self-loop Tl = {tl.1 , tl.2 , · · · , tl.k } and without self-loop TO =
{tO.1 , tO.2 , · · · , tO.k },Tl and TO can be shrunk into a single transition, where
input arc α0 and new output arc β 0 are defined by the following formulas;
α0 = (sO.1 · αO.1 + sO.2 · αO.2 + · · · + sl.k2 · αl.k2 )
−(sl.1 · βl.1 + sl.2 · βl.2 + · · · + sl.k2 · βl.k2 ) /sO.1 (11)
β10 = sO.1 · β1
β20 = sO.2 · β2
..
.
0
βk2 = sl.k2 · βk2 (12)
Note that, in equation (11) α0 > 0 should be held.
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
94 A Mizuta, QW Ge and H Matsuno
Fig. 8. Self-loop structure
4 Dependent Shrink Algorithm and an Example
In this section, we propose a dependent shrink algorithm based on the dependent
shrink method. Furthermore, we apply this algorithm to IL-3 signaling pathway
Petri net model (shown in Fig. 10), which is transformed from IL-3 phenomenon
model (shown in Fig. 9) obtained from the website [10]. Note that IL-3 is a
glycoprotein and is known to be involved in the immune response [6–9].
4.1 Outline of shrink process
The shrink process of dependent subnet can be briefly described as follows:
step1: Shrink of self loop structure
A place randomly selected from a Petri net is stored in a queue after the
conversion of the self-loops and the structures of conflict of it.
step2: Shrink of conflict structure
If a place picked up from the queue has a self-loop or a transition of one-input
and one-output, this place is shrunk.
step3: Changing weight of the input arc
If shrunk Petri net has a multiple input place, it re-stores to the queue,
performing the above step2 again. This procedure is repeated until the queue
becomes empty.
The variables used in the algorithm are as follows:
– P N0 is a given signaling pathway Petri net model constituted by T0 , P0 , and E0 .
– N is a variable that stores Petri net after dependent shrink, constituted by
T, P, andE.
– Q is a queue.
– X is a set of place initially set as a given place set P0 .
– f is a flag, by which dependent shrink pattern is determined.
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
Dependent shrink for signaling pathways 95
Fig. 9. IL-3 phenomenon model
4.2 Dependent shrink algorithm
The following algorithm is used to shrink dependent subnets into a single tran-
sition in order to find transitions with interdependent firing frequency.
Algorithm: Dependent shrink
Input: P N0 = (T0 , P0 , E0 )
Output: Shrunk Petri net N = (T, P, E)
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015
96 A Mizuta, QW Ge and H Matsuno
Main(P N0 )
1◦ T ← T0 , P ← P0 , E ← E0 , N ← (T, P, E)
2◦ X ← P , Q ← φ
3◦ while (X 6= φ)
Pull an element x from X(X ← X − {x})
Enqueue(Q, x)
Shrink1(N, x)
4◦ Shrink2(N, Q)
Shrink1(N, x)
1◦ if (|• x ∩ x• | ≥ 1) then
f ←1
Arcweight(N, x, f )
2◦ if (|x• | ≥ 2) then
f ←2
Arcweight(N, x, f )
Shrink2(N, Q)
1◦ while (|Q| ≥ 1)
x ← Dequeue(Q)
if (|• x ∩ x• | ≥ 1) then
f ←1
Enqueue(Q, x)
else if (|• x| = |x• | = 1) then
f ←3
else if (|• x| ≥ 2) then
f ←4
Enqueue(Q, x)
if (f 6= 4) then
Arcweight(N, x, f )
Arcweight(N, x, f )
1◦ if (f = 1) then
∀t0 ∈ • x ∩ x•
α(x, t0 ) = α(x, t0 ) − β(t0 , x)
if (α(x, t0 ) < 0) then
β(t0 , x) = |α(x, t0 )|
E ← E − {(x, t0 )}
else if (α(x, t0 ) > 0) then
E ← E − {(t0 , x)}
else if (α(x, t0 ) = 0) then
E ← E − {(t0 , x), (x, t0 )}
◦
2 else if (f = 2) then
T ← T ∪ {t0 }
E ← E ∪ {(x, t0 )} ∪ {(u, t0 )|u ∈ • z, z ∈ x• } ∪ {(t0 , v)|v ∈ z • , z ∈ x• }
Choose z 0 ∈ x• .
∀z ∈ x• − {t0 }
α(x, t0 ) = α(x, t0 ) + s(z) ∗ α(x, z)
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Dependent shrink for signaling pathways 97
∀v ∈ z • , z ∈ x•
β(t0 , v) = s(z) ∗ β(z, v)
∀u ∈ • z, z ∈ x•
α(u, t0 ) = s(z) ∗ α(u, z)/s(z 0 )
α(x, t ) = α(x, t0 )/s(z 0 )
0
T ← T − {z|z ∈ x• − {t0 }}
◦
3 else if (f = 3) then
T ← T ∪ {t0 }
Let zi , zo be {zi } = • x, {zo } = x• (due to |• x| = |x• | = 1).
E ← E ∪ {(u, t0 )|u ∈ • zi ∪ • zo } ∪ {(t0 , v)|v ∈ zi• ∪ zo• }
∀u ∈ • zi
α(u, t0 ) = α(u, zi )
∀u ∈ zi •
β(t0 , u) = β(zi , u)
∀v ∈ • zo
α(v, t0 ) = β(zi , x) ∗ α(v, zo )/α(x, zo )
∀v ∈ zo •
β(t0 , v) = β(zi , x) ∗ β(zo , v)/α(x, zo )
T ← T − {zi |zi ∈ • x} − {zo |zo ∈ x• }
P ← X − {x}
When the above algorithm is applied, a dependent subnet, say S, is trans-
formed into a single transition, say tS . Obviously S and tS possess the same
input and output places. As the result, for each input place, p, the tokens flowed
out of p per unit time are the same before and after dependent shrink. Similarly,
the tokens flowed into an output place per unit time are the same.
4.3 An Example for Signaling Pathway Petri Net Model
Here we give an example to show an application of our proposed algorithm.
The algorithm is applied to dependent shrink for IL-3 Petri net model (see
Fig.10 (a)), obtained from the website [10]. As the result, the original IL-3 Petri
net model shown in Fig.10 (a) is shrunk into Fig.10 (c). This means that the
firing frequency of all transition in Fig.10 (a) are denpendent each other. In the
intermediate shrunk net (see Fig. 10 (b)), by assuming the firing frequency of
the input transition fI be 1, the weights of input and output arcs be 12 and 1,
respectively, then the firing frequency of output transition fO is 12 . In this way,
we can calculate all of the firing frequency of transitions in the IL-3 Petri net
model from one firing frequency in this model.
5 Conclusion
In this paper, after giving basic definitions of Petri net and modeling method,
we introduced dependent shrink method and its properties to find dependent
subnet. Further, we designed an algorithm of dependent shrink and applied it to
IL-3 signaling pathway Petri net model as an example.
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98 A Mizuta, QW Ge and H Matsuno
Fig. 10. Shrunk IL-3 Petri net model
By applying the dependent shrink algorithm, IL-3 Petri net model is con-
verted to a simple model, with which we could find the transitions which are
dependent each other.
This algorithm allows us to obtain firing frequencies of all transitions in a
dependent subnet only by measuring reactions corresponding to the transitions
by biological experiments in the simple model.
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Dependent shrink for signaling pathways 99
In this paper, we only discussed discrete Petri nets. By extending transitions
to include firing speed, it is possible to extend our method to continuous Petri
nets.
As a future work, we need to improve our algorithm so that it can indicate
transitions corresponding to measurable reactions by biological experiment. Also
the uniqueness of our algorithm needs to be investigated.
Acknowledgement
We would like to thank Dr. Adrien Fauré at Yamaguchi University for his support
in proofreading the manuscript. This work was partially supported by Grant-in-
Aid for Scientific Research (B) (23300110) from Japan Society for the Promotion
of Science.
References
1. C. Li, S. Suzuki, Q.W. Ge, M. Nakata, H. Matsuno, S. Miyano, “Structural mod-
eling and analysis of signaling pathways based on Petri net”, Journal of Bioinfor-
matics and Computational Biology, Vol.4, No.5, pp.1119-1140, 2006.
2. Y. Miwa, Y. Murakami, Q.W. Ge, C. Li, H. Matsuno, S. Miyano, “Delay time
determination for the timed Petri net model of a signaling pathway based on its
structural information”, IEICE on Fundamentals of Electronics, Communications
and Computer Sciences, Vol.E93-A, No12, pp2717-2729, 2010 (in Japanese).
3. Y. Murkami, Q.W. Ge, H. Matsuno, “Consideration on the token retention-free in
timed Petri net model based on the signaling pathway characteristics”, Technical
Report of IEICE, Vol.111, No.453, MSS2011-77, pp.29-34, 2012 (in Japanese).
4. T. Matsumoto, Q.W. Ge, H. Matsuno, “An algorithm for finding dependent subnets
in a retention-free Petri net,” Technical Report of IEICE, MSS2012-50, pp.27-32,
2013 (in Japanese).
5. J.L. Peterson, Petri net theory and the modeling of systems, Prentice Hall, 1981.
6. K. Sugamura, K. Miyazono, K. Miyazawa, N. Tanaka, Saitokain zoushokuinshi
yougo raiburari (Term Library of Cytokines and Growth factor), Yodosha, 2005 (in
Japanese).
7. W. Chen, M.O. Daines, G.K. Khurana Hershey, “Turning off signal transducer and
activator of transcription (STAT): the negative regulation of STAT signaling,” The
Journal of allergy and clinical immunology, 114(3), pp.476-489, quiz 490, 2004.
8. T. Kisseleva, S. Bhattacharya, J. Braunstein, C. W. Schindler, “Signaling through
the JAK/STAT pathway,” recent advances and future challenges, Gene, 285(1-2),
pp.1-24, 2002.
9. R.P. de Groot, P.J. Coffer, L. Koenderman, “Regulation of proliferation, differenti-
ation and survival by the IL-3/IL-5/GM-CSF receptor family,” Cellular signalling,
10(9), pp.619-628, 1998.
10. Petri Net Pathways, http://genome.ib.sci.yamaguchi-u.ac.jp/pnp/(accessed
April 20th)
Proc. BioPPN 2015, a satellite event of PETRI NETS 2015