=Paper= {{Paper |id=None |storemode=property |title=A Hybrid Petri Net Model of the Eukaryotic Cell Cycle |pdfUrl=https://ceur-ws.org/Vol-852/paper5.pdf |volume=Vol-852 }} ==A Hybrid Petri Net Model of the Eukaryotic Cell Cycle== https://ceur-ws.org/Vol-852/paper5.pdf

Proc. BioPPN 2012, a satellite event of PETRI NET 2012

A Hybrid Petri Net Model of
the Eukaryotic Cell Cycle

Mostafa Herajy and Martin Schwarick

Brandenburg University of Technology at Cottbus,
Computer Science Institute,
Data Structures and Software Dependability,
Postbox 10 13 44, 03044 Cottbus, Germany
http://www-dssz.informatik.tu-cottbus.de/

Abstract. System level understanding of the repetitive cycle of cell
growth and division is crucial for disclosing many unexpected princi-
ples of biological organisms. The deterministic or stochastic approach
are alone not sufficient to study such cell regulation due to the complex
reaction network and the existence of reactions with different time scales.
Thus, integration of both approaches is necessary to study such biochem-
ical networks. In this paper we present a hybrid Petri net model to study
the eukaryotic cell cycle using Generalised Hybrid Petri Nets. The pro-
posed model is intuitively and graphically represented through Petri net
primitives. Moreover, it can capture intrinsic and extrinsic noises and
deploys stochastic as well as deterministic reactions. Additionally, self-
modifying weights are motivated and introduced to Snoopy – a tool for
animating and simulating Petri nets.

Keywords: Generalised hybrid Petri nets; hybrid modelling; eukaryotic
cell cycle

1 Introduction
The reproduction of eukaryotic cells is controlled by a complex regulatory net-
work of reactions known as cell cycle [17,18,21]. Through it, cells grow, replicate
and divide into two daughter cells [12,19]. This regulation cycle consists of four
phases: S phase (synthesis) and M phase (mitosis) separated by two gap phases:
G1 and G2 [21]. During the S phase, the cell replicates all of its components,
while it divides each component more or less evenly between the two daughter
cells at the end of the M phase [12]. After the S phase, there is another gap (G2)
where the cell ensures that the duplication of DNA has completed and prepares
itself for mitosis. Newborn cells are not replicated and located at the G1 gap.
Furthermore, the processes of synthesis and mitosis alternate with each other
during the reproduction process. Understanding such control cycles is crucial
for revealing defects in cell growth which underlies many human diseases (e.g.,
cancer) [22].
In the eukaryotic cell cycle, the alternation between the S and the M phase as
well as the balance of growth and division is governed by the activity of a family

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

of cyclin-dependent protein kinases (CDK) [12]. Therefore, many computational
models have been constructed to study the control system of CDK (e.g., in
[1,12,17,18,21]). Some of these models are based on the deterministic approach
which represents changes of species concentrations as continuous variables that
evolve deterministically and continuously with respect to time. However, such
approach does not capture the variability of cell size due to the fluctuation of
some species which usually exist in low numbers of molecules [4]. Motivated by
this argument, a number of stochastic models have been created and simulated
using either a stochastic simulation algorithm (e.g., [12]) or by introducing noise
to the model through Langevin equation [20]. However, the stochastic approach
is computationally expensive, particularly when the model under study contains
reactions of high rates or species of large numbers of molecules.
Similarly, the eukaryotic cell cycle model exhibits high reaction rates of some
reactions while some other reactions have low rates. The latter types are respon-
sible for the intrinsic noise due to molecular fluctuations [14]. The existence of
reactions of different time scales (fast and slow) suggests the simulation using
a hybrid approach. In [14] and [19] two different hybrid approaches are used to
simulate the progression of cell cycle.
Correspondingly, Generalised Hybrid Petri Nets (GHP Nbio ) have been in-
troduced, in [10] and [11], to represent and simulate stiff biochemical networks
where fast reactions are represented and simulated continuously, while slow re-
actions are carried out stochastically. GHP Nbio provide rich modelling and sim-
ulation functionalities by combining all features of Continuous Petri Nets [2] and
Extended Stochastic Petri Nets [15], including three types of deterministic tran-
sitions. Moreover, the partitioning of the reaction networks can either be done
off-line before the simulation starts or on-line while the simulation is in progress.
The implementation of GHP Nbio is available as part of Snoopy [8] - a tool to
design and animate or simulate hierarchical graphs, among them qualitative,
stochastic, continuous and hybrid Petri nets. Indeed, the cell cycle model is an
ideal case where the majority of GHP Nbio features can be demonstrated.
In this paper we present another argument to motivate the hybrid simulation
of the cell cycle control system. The cell cycle model contains some components
which would be better represented as continuous processes (e.g., volume growth),
while other reactions of low rates are vital to represent them as stochastic pro-
cesses. For instance, Mura and Csikasz-Nagy constructed in [17] a stochastic
version of the model in [1] using stochastic Petri nets. However, they got stuck
with the problem of representing cell growth processes which evolve continuously
and exponentially with respect to time using stochastic Petri net primitives. In-
deed cell growth is a typical example where continuous transitions could be used.
Moreover, our proposed model is graphically and intuitively represented in terms
of Petri nets.
The paper is organised as follows: we start off by a brief introduction of
Generalised Hybrid Petri Nets. After that, some related work is pinpointed.
Next, we present our hybrid Petri net model of the eukaryotic cell cycle and
describe in detail some of its key modelling components. In Section 5 we show

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

the simulation results produced by Snoopy’s hybrid simulation engine. Finally,
we sum up with conclusions and outlook.

2 Related Work

Mura and Csikasz-Nagy constructed in [17] a stochastic Petri net model based
on the work of [1] to study the effect of noise on cell cycle progression. However,
some components could not intuitively be represented using SPN primitives only
(e.g., cell growth). Moreover, their model is based on phenomenological rate laws
(e.g., Michaelis-Menten) which do not work well with stochastic simulation algo-
rithms [12]. Sabouri-Ghomi et al. [18], and Kar et al. [12], asserted that applying
Gillespie’s stochastic simulation algorithm [3] directly to phenomenological rate
laws might produce incorrect results. Therefore, they unpacked the phenomeno-
logical deterministic model of Tyson-Novak [21] in terms of elementary mass
action kinetics. The Tyson-Novak model is based on a bistable switch between
the complex CycB-Cdk1 (denoted by variable X) and the complex Cdh1-APC
(denoted by the variable Y). CycB-Cdk1 phosphorylates Cdh1-APC and free
Cdh1-APC catalyses the degradation of CycB-Cdk1. To model a complete cell
cycle, Kar et al. [12] unpacked the effect of Cdc20 and Cdc14 which are lumped in
the variable Z in the Tyson-Novak model. High activity of CycB-Cdk1 promotes
the synthesis of Cdc20 which activates Cdc14. Finally the dephosphorylated
Cdc14 activates Cdh1-APC. The Kar et al. model accounts for both intrinsic
and extrinsic noises. Intrinsic noise is due to the fluctuation of species of low
numbers of molecules, while extrinsic noise is due to the unequal division of the
cell between the two daughter cells [12].
In [19], a hybrid model which combines ordinary differential equations (ODEs)
and discrete Boolean networks has been constructed to adapt quantitative as
well as qualitative parts in the same model. The latter approach requires less
knowledge about realistic kinetic rate constants. Liu et al. [14] simulate the
stochastic model of [12] by statically partition the model reactions into slow and
fast ones. However, as they have reported in their paper, the resulting hybrid
model requires more simulation time than the original stochastic one. Therefore,
they have (re)packed fast reactions in terms of phenomenological rate lows. The
model presented in this paper differs from the Liu et al. one in the intuitive
graphical representation and execution time of the hybrid model.
In this paper a hybrid Petri net model of the eukaryotic cell cycle is pre-
sented. The model is hybrid in the sense that it combines continuous, stochas-
tic and immediate transitions to represent deterministic, stochastic and control
components. Our main goal is to show how such class of models are intuitively
represented and executed using hybrid Petri net primitives. Using Snoopy’s sim-
ulator, it can be simulated either deterministically, stochastically or in a hybrid
way.

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

3 Generalised Hybrid Petri Nets
To model stiff biochemical networks, GHP Nbio [11] combine both stochastic and
continuous elements in one and the same model. Indeed, continuous and stochas-
tic Petri nets complement each other. The fluctuation and discreteness can be
conveniently modelled using stochastic simulation and at the same time, the
computationally expensive parts can be simulated deterministically using ODE
solvers. Modelling and simulation of stiff biochemical networks are outstanding
functionalities that GHP Nbio provide for systems biology.
Generally speaking, biochemical systems can involve reactions from more
than one type of biological networks, for example gene regulation, metabolic
pathways or signal transduction pathways. Incorporating reactions which belong
to distinct (biological) networks, tends to result into stiff systems. This follows
from the fact that gene regulation networks species may contain a few number
of molecules, while metabolic networks species may contain a large number of
molecules [13].
In the rest of this section, we will give a brief introduction of GHP Nbio in
terms of the graphical representation of its elements as well as the firing rules
and connectivity between continuous and stochastic net parts.

3.1 Elements
The GHP Nbio elements are classified into three categories: places, transitions
and arcs.
GHP Nbio offer two types of places: discrete and continuous. Discrete places
(single line circle) hold non-negative integer numbers which represent e.g., the
number of molecules of a given species (tokens in Petri net notions). On the
other hand, continuous places - which are represented by shaded line circle -
hold non-negative real numbers which represent the concentration of a certain
species.
Furthermore, GHP Nbio offer five transition types: stochastic, immediate,
deterministically delayed, scheduled, and continuous transitions [7]. Stochastic
transitions which are drawn in Snoopy as a square, fire with an exponentially dis-
tributed random delay. The user can specify a set of firing rate functions, which
determine the random firing delay. The transitions’ pre-places can be used to
define the firing rate functions of stochastic transitions. Immediate transitions
(black bar) fire with zero delay, and have always highest priority in the case
of conflicts with other transitions. They may carry weights which specify the
relative firing frequency in the case of conflicts between immediate transitions.
Deterministically delayed transitions (represented as black squares) fire after a
specified constant time delay. Scheduled transitions (grey squares) fire at user-
specified absolute time points. Continuous transitions (shaded line square) fire
continuously in the same way like in continuous Petri nets. Their semantics are
governed by ODEs which define the change in the transitions’ pre- and post-
places. More details about the biochemical interpretation of deterministically
delayed, scheduled, and immediate transitions can be found in [9] and [15]. To

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

d1 d1

P/2

P T d2 P T d2
P/2

(a) (b)

Fig. 1. Self-modifying weight illustrated by a simple biological example. (a) cell division
cannot be modelled (b) cell division can intuitively be modelled.

simplify the presentation, we occasionally refer to stochastic, immediate, deter-
ministically delayed or scheduled transitions as discrete transitions.
The connection between those two types of nodes (places and transitions)
takes place using a set of different arcs (edges). GHP Nbio offer six types of
arcs: standard, inhibitor, read, equal, reset and modifier arcs. Standard arcs
connect transitions with places or vice versa. They can be discrete, i.e., carry
non-negative integer-valued weights (stoichiometry in the biochemical context),
or continuous i.e., carry non-negative real-valued weights. In addition to their
influence on the enabling of transitions, they affect also the place marking when
a transition fires by adding (removing) tokens from the transition’s post-places
(pre-places). For more details see [10].
To support the special modelling requirements of some biological models (e.g.,
cell cycle model), arc weights are allowed to be a pre-place of a transition [23]
or even a function which is defined in terms of the transition’s pre-places [16].
Consider the following simple biological example. When a cell divides the
mass between two daughter cells, each daughter takes approximately half of the
mass. This example cannot be modelled using standard Petri nets as shown in
Figure 1a. In Figure 1b, Using self-modifying weight; the ongoing arc of the
transition ”t” has weight equal to the marking of the place ”P”, while each of
the two outgoing arcs has weight equal to the half marking of place ”P”.
Motivated by the case study of this paper, self-modifying weights are intro-
duced to all arc types supported by Snoopy (standard, read, inhibitor, and equal
arc). For more detail see Section 4.2.
Extended arcs like inhibitor, read, equal, reset, and modifier arcs can only be
used to connect places to transitions, but not vice versa. A transition connected
with an inhibitor arc is enabled if the marking of the pre-place is less than the
arc weight. Contrary, a transition connected with a read arc is enabled if the
marking of the pre-place is greater than or equal to the arc weight. Similarly, a
transition connected using an equal arc is enabled if the marking of the pre-place
is equal to the arc weight.

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

Places

Discrete Continuous

Transitions

<1> [_SimStart,1,_SimEnd]

Stochastic Continuous Immediate Deterministic Scheduled

Edges

Standard Read Inhibitor Equal Reset Modifier

Fig. 2. Graphical representation of the GHP Nbio elements. Places are classified as
discrete and continuous, transitions as continuous, stochastic, immediate, determinis-
tically delayed, and scheduled, and edges as standard, inhibitor, read, equal, reset, and
modifier.

The other two remaining arcs do not affect the enabling of transitions. A reset
arc is used to reset a place marking to zero when the corresponding transition
fires. Modifier arcs permit to include any place in the transitions’ rate functions
and simultaneously preserve the net structure restriction.
The connection rules and their underlying formal semantics are discussed
in more details below. Figure 2 provides a graphical illustration of all elements.
Although this graphical notation is the default one, they can be easily customised
using the Petri nets editing tool, Snoopy.

3.2 Connection Rules

A critical question arises when considering the combination of discrete and con-
tinuous elements: how are these two different parts connected with each other?
Figure 3 provides a graphical illustration of how the connection between different
elements of GHP Nbio takes place.
Firstly, we will consider the connection between continuous transitions and
the other elements of GHP Nbio . Continuous transitions can be connected with
continuous places in both directions using continuous arcs (i.e., arc with real-
valued weight). This means that continuous places can be pre- or post-places
of continuous transitions. These connections typically represent deterministic
biological interactions.

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

or or

Continuous Transition
Discrete Transition

Fig. 3. Possible connections between GHP Nbio elements. The restrictions are: discrete
places can not be connected with continuous transitions using standard arcs, continuous
places can not be tested with equal arcs, and continuous transitions can not use reset
arcs.

Continuous transitions can also be connected with discrete places, but only
by one of the extended arcs (inhibitor, read, equal, and modifier). This type of
connection allows a link between discrete and continuous parts of the biochemical
model.
Discrete places are not allowed to be connected with continuous transitions
using standard arcs, because the firing of continuous transitions is governed by
ODEs which require real values in the pre- and post-places. Hence, this cannot
take place in the discrete world.
Secondly, discrete transitions can be connected with discrete or continuous
places in both directions using standard arcs. However, the arc’s weight needs to
be considered. The connection between discrete transitions and discrete places
takes place using arcs with non-negative integer numbers, while the connection
between continuous places and discrete transitions is weighted by non-negative
real numbers. The general rule to determine the weight type of arcs is the type
of the connected place.

4 The Model

Figure 4 shows the hybrid Petri net model based on the previous one introduced
by Kar et al. in [12]. Proteins, genes and mRNAs are represented by places.
Transitions represent reactions. We use the same kinetic parameters and initial
values. For the sake of compactness will not repeat them again here. Initial mark-
ings are shown inside the places. Moreover, we use Snoopy’s logical node features
to simplify connections between different nodes. For example, place X and Y
are involved in many reactions which decreases the network’s readability. We
repeat them multiple times with same names to keep the model understandable

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

(logical places). Likewise the transition ”divide” (logical transitions). Further-
more, the increase of cell volume size is intuitively represented using a continuous
transition with a rate µ · V , where µ is the growth factor and V is the cellular
volume.
The model contains three different transition types: continuous, stochastic,
and immediate. Continuous transitions simulate the corresponding reactions de-
terministically, while stochastic transitions carry them out stochastically. The
latter transitions are responsible for molecular fluctuations. Immediate transi-
tions monitor the model evolution and perform the division when the free number
of molecules of Cdh1 APC reaches a certain threshold (Ŷ = Y + Y X + XY ).
In the sequel we present in more detail some of the model’s key components
and the corresponding GHP Nbio representations.

4.1 Decision to Perform Division

When the number of molecules of Ŷ becomes greater than a certain threshold
(in our case 1200), the cell can divide the mass and other components between
the two daughter cells. In Figure 5, this process is represented by an immediate
transition ”check” with the weight Ŷ > threshold. Recall that immediate transi-
tion weights determine the firing frequencies of immediate transitions in case of
conflicts. A weight of zero means that a transition cannot fire at all. Therefore,
when the transition ”check” has weight greater than zero, it adds a token to the
place ”ready to divide” which signals the transition ”divide” to carry out the
division. To give the transition ”divide” a chance to fire before re-checking the
value of Ŷ , an inhibitor arc is used to constrain this case.
An interesting characteristic of the model is the division process. Although
the division can take place when the value of Ŷ is greater than a certain thresh-
old, it does not do that all the times. For example, at the beginning of the
simulation, the initial value of Ŷ satisfies the dividing criterion. However; the
cell should not divide because it is still at G1 phase which means that it has to
replicate before it can divide. We model these cases by adding a new immedi-
ate transition which detects the critical value of Ŷ , before checking for division.
Therefore the transition ”critical” monitors the value of Ŷ . When the value of
Ŷ goes below a certain threshold, it enables the division process.

4.2 Cell Division and Self-modifying Weights

When a cell divides, it divides all of its components more-or-less evenly between
two daughter cells. This is another ideal case to demonstrate self-modifying
weights [23]. In Figure 5, when the transition ”divide” fires, it removes all of the
current marking of the place V and adds V /2 to it. To permit uneven division
of the cell volume and other components, arc weights can be a function which
operates on the current place marking [16]. However, we restrict the places used
in arc weights to the transitions’ pre-places to maintain the Petri net structure.

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

ready_for_check
X_Y X_Y mRNA_y
divide mRNA_y
36 divide
X_Y
X_Y/2 G_Y
mRNA_y/2 36

232 Y
Yp Yp 2189
21 V
232 2189 critical check ready_for_divide
Y_X
Yp Z Yp Y 83
232 9 Yp/2 divide

Y
2189 divide V/2 V
Yp_X
divide Z
21
Z_Yp 9 Z
Yp_X/2
127 Z/2
V V

Z_Yp 133 21
X Yp_X
Z_Yp/2
6

divide

divide 21 divide
Y

X_Y
Y/2 TF G_TF
2189 36
Y R6 TF/2
divide mRNA_z
divide H
divide
1120 2799
H
Y_X/2 Y_X H/2 TF mRNA_tf mRNA_z/2 mRNA_z

V 21
83 6
X
Y_X
divide mRNA_h mRNA_h
6
V 21
X/2 X C

mRNA_h/2
X G
divide

G_GH C
mRNA_x/2 TF_p TF_p2
mRNA_x 2
1
TF_p
divide
mRNA_x 2 21
TF_p2
21 TF_p/2 V
V TF_p2/2
G_X TF_p2

Fig. 4. A Generalised Hybrid Petri Nets representation of the eukaryotic cell cycle. The
model employs different types of transitions: continuous, stochastic and immediate. All
reactions affecting m RNAs are represented and simulated stochastically. Repetitive
nodes (places and transitions) with same names are logical nodes. When the transition
”divide” fires, it divides the current place marking more or less equally. The type of
division (equal, or unequal) depends on the outgoing arc weight and its effect takes
place using self-modifying weight.

4.3 Transition Partitioning

The model in Figure 4 contains transitions which fire at different rates. For in-
stance, transition ”R3”, as illustrated in Figure 6a, fires more frequently than
”R1”. Slow transitions should be simulated stochastically to account for molec-
ular fluctuations, while fast transitions need to be simulated continuously to

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

ready_for_check

X_Y

Y
divide
2189

critical check ready_for_divide
Y_X
V/2 V
83

Fig. 5. A sub-net for modelling the decision of the division process. The transition
”critical” monitors the value of Ŷ and adds a token to ”ready for check” when Ŷ < 300.
Later, when the value of Ŷ increases and becomes greater than a threshold (1200),
the transition ”check” fires and adds a token to ”ready for divide” which signals the
transition ”divide” to perform the division. Inhibitor arcs are used as a check point for
the sequence of events: critical → check → divide.

increase the numerical efficiency. Indeed, the latter types consume the majority
of computational resources.
In this model, transitions are partitioned statically before the simulation
starts. The transition type is decided by executing a single run and analyse the
results as in Figure 6. Increasing (decreasing) the accuracy of the model re-
sults involves converting more continuous (stochastic) transitions into stochastic
(continuous) ones. Similarly, controlling the speed of the model simulation will
require the opposite procedure.
Another approach to do the partitioning is to perform it dynamically during
the simulation. Using this technique, a transition changes its type from stochastic
to continuous or vice versa according to the current firing rate. GHP Nbio provide
the user with a trade-off between efficiency and accuracy by permitting the user
to specify two thresholds: a0min and a0max , the minimum and maximum cumu-
lative propensity, respectively. Moreover, two other thresholds are required to
perform dynamic partitioning: the place marking threshold and the transition
rate threshold. The former is used to ensure that species concentrations are large
enough to be simulated continuously, while the latter is used to partition tran-
sitions into fast and slow based on their rates. For a transition to be simulated
continuously its rate has to exceed the rate threshold and the marking of all its
pre-place must be greater than the marking threshold.
Nevertheless, in both cases cell growth has to be represented and simulated
continuously. Using off-line partitioning, this can be easily told to the simulator
by drawing a continuous transition. However, in the case of dynamic partitioning;

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Proc. BioPPN 2012, a satellite event of PETRI NET 2012

6000
R1 R26
R3 R18
8000

5000
7000

6000 4000

5000
Rates

Rates
3000
4000

3000 2000

2000
1000
1000

0 0
0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
time time

(a) (b)

Fig. 6. Example of different transition firing rates. (a) transition ”R3: X+Y → Y X”
fires more frequently than transition ”R1: mRNA x → mRNA x+X” and (b) transition
”R18: H → H+TF” fires much more than ”R26: mRNA tf→ mRNA tf +TF”.

the transition rate threshold should be set less than the expected rate of cell
growth.

5 Simulation Results
In this section, we compare the simulation results of the following scenarios:
when all of the reactions are simulated stochastically, when reactions related
only to mRNAs are simulated stochastically and when all reactions are simulated
continuously. In all cases cell growth is simulated using a continuous transition.
Figure 7 shows the results of the three approaches using species with low numbers
of molecules (mRNA x and mRNA z). Since reactions related to mRNAs are
simulated stochastically in hybrid and stochastic simulations, their results are
close to each other.
Figure 8 shows time course simulation results of proteins X and Y. In hybrid
and stochastic simulations, X and Y are affected with fluctuations of mRNAs.
while in continuous one there is no such effect.
Figure 9 compares continuous and hybrid simulation results of the volume size
(V). Using continuous simulation, cell divides all the time equal and the model
produces no variability in its volume size. The hybrid simulation shows variability
in the volume size because species of low numbers of molecules (e.g., mRNAs)
are simulated stochastically which account for the molecular fluctuations and
therefore, they are responsible for the intrinsic noise [12].
As a conclusion, the hybrid simulation approach can reproduce the results of
the stochastic approach. However, substantially amount of simulation time could
be saved. Fortunately, the resulting system of ordinary differential equations
(ODEs) of this model is not stiff (for more details, see [5] and [6]). Therefore,
an explicit ODE solver can be used to increase the performance of the hybrid
simulation engine in connection with the stochastic simulation algorithms (SSA).

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18 12
stochastic stochastic
hybrid hybrid
continuous continuous
16
10
14

12 8
number of molecules

number of molecules
10
6
8

6 4

4
2
2

0 0
0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
time time

(a) (b)

Fig. 7. Time course result of the model in Figure 4 using Snoopy simulator (a) mRNA x
and (b) mRNA z.

2500 stochastic 3000 stochastic
hybrid hybrid
continuous continuous

2500
2000

2000
number of molecules

number of molecules

1500

1000
1000

500
500

0 0
0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
time time

(a) (b)

Fig. 8. Time course result of species with large number of molecules; (a) Y and (b) X.

35
V 35 V

30
30

25
25

20
20
size

size

15 15

10 10

5 5

0 0
500 600 700 800 900 1000 500 600 700 800 900 1000
time time

(a) (b)

Fig. 9. Continuous and hybrid simulation results for the cellular volume (V); (a) con-
tinuous result and (b) hybrid simulation result.

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6 Conclusions and Outlook

In this paper we have presented a hybrid Petri net model of the eukaryotic
cell cycle. The model can be executed using either continuous, stochastic or
hybrid simulators. It employs continuous, stochastic and immediate transitions
to intuitively represent the entire model logic.
The model is implemented using Snoopy which is available free of charge at
http://www-dssz.informatik.tu-cottbus.de/snoopy.html.
Self-modifying weight is a new added feature to Snoopy which is currently
not available in the official Snoopy release.
From the simulation results we notice that hybrid simulation produces results
close to the stochastic one while simulation efficiency could be preserved. Indeed,
the reactions of this model could easily be separated into slow and fast reactions,
which makes it an ideal case study for hybrid simulation algorithms.
Self-modifying arcs are of paramount importance to model such biological
cases since they provide a direct tool to program some biological phenomenon
(e.g., cell division). Therefore, we intend to add more functionalities into this
direction to permit more user-defined operators depending on transition’s pre-
places.
So far the partitioning of the reactions into stochastic and deterministic ones
is carried out using a heuristic approach (see Section 4.3). However, as it has been
risen during the review process; a better justification for the partitioning could
be performed. For instance, the fast processes can be regarded as processes that
could be described by quasi (or pseudo)-steady state approach, assuming that
they reach equilibrium rapidly. In other words, they could be better described
by setting the corresponding ODE to zero and solving for the fast variables.
In contrast, continuous dynamics could be seen as more appropriate for abun-
dant molecules whose concentration display a small coefficient of variation, and
stochastic dynamics for those molecules evolving at low copy number.
The presented model could be viewed as a sub-net in a bigger network of
reactions (e.g., modelling budding yeast cell cycle or Fission yeast cells). Snoopy’s
hierarchical nodes might simplify such task as they provide an easy tool to insert
a sub net in a bigger one.

7 Acknowledgements

The authors acknowledge the comments of Monika Heiner during the implemen-
tation of GHP Nbio and the preparation of this manuscript.

References

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