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肥ICE TRANS．FUNDAMENTALS， VOLE87－A， NO．11NOVEMBER 20042919
PAPER
Special Section on Concurrent Systems and Hybrid SystemsModeling and Simulation of Fission Yeast Cell Cycle on Hybrid
Functional Petri Net
Sachie FUJITAt， Mika MATSUIt， Hiroshi MATSUNOt＊a）， and Satoru MIYANO＊＊t， Members
SUMMARY Through many researches on modeling and analyzing bi－
ological pathways， Peui net has recognized as a prornising method for rep－
resenting biological pathways． Recently， Matsuno et al． （2003） introduced
hybrid functional Petri net （HFPN） for giving more intuitive and natural
biological pathway modeling method than existing Petri nets． They also
developed Genomic Object Net （GON） which employs the HErpN as a ba－
sic舳itecture． Many knds of biological pa血ways have been modeled
with the HFPN and simulated by the GON． This paper gives a new HITPN
model of “cell cycle of fission yeast” with giving six basic HFPN compo－
nents of tYpical biological reactions， and demonstrating the method how
biological pathways can be modeled with these HFPN components． Simu－
lation results by GON suggest a new hypothesis which will help biologist
for performing further expetments．
勧，｝ジ。鳩’励rid fanctional Petri●neちGenomic Object Net， b’0109’cα1
pathways， cell cycle， simulation
1． lntroduction
Petri net ［30］ is a description method for modeling concur－
rent systems mainly used so far to model artificial systems
such as m舳ct曲g systems［28］and communicationprotocols ［35］． The first attempt to use Petri net for mod－
eling biological pathways was made by Reddy et al． ［29］
which gave a method of representation of metabolic path－
ways． Hofestadt expanded this method to model metabolic
networks ［12］． After their works， several enhanced Peni
nets have been used for modeling the behaviors of biologi－
cal phenomena， Genrich et al． ［7］ modeled metabolic path－
ways with the colored Petri net with assigning enzymatic
reaction speeds to the transitions， and simulated a chain of
these reactions quantitatively． Voss et al． used the colored
Petri net in different way， accomplishng a qualitative analy－
sis of steady state in metabolic pathways ［34］． The stochas－
tic Petri net has been applied to model a variety of biological
pathways；the ColE l plas血d replication［9］，the response of
a32 transcription factor to a heat shock ［31］， and the inter－
action knetics of a viral invasion［32］． On血e o血er hand，
we have shown that the gene regulatory network of 1 phage
can be more naturally modeled as a hybrid system of “dis－
crete” and “continuous” dynamics ［15］ by employing hy一
Manuscript received March 24， 2004．
Manuscript revised June 15， 2004．
Final manuscript received July 9， 2004．
tThe authors are with Graduate School of Science and Engi－
neept． ng， Yamaguchi University， Ube－shi， 755－861 1 Japan．
ttThe author is with Faculty of Science， Yamaguchi University，
Yamaguchi－shi， 753－8512 Japan．
tttThe author is with Human Genome Center， lnstitute of Medi－
cal Science， ’IThe University of Tokyo， Tokyo， 108－8639 Japan．
a） E－mai1： matsuno＠sci．yamaguchi－u．ac．jp
brid Petri net （HPN） architecture ［2］， ［5］． lt is also observed
［8］ that biological pathways can be handled as hybrid sys－
tems， e．g． protein concentration dynamics behaves comin－
uously being coupled with discrete switches； protein pro－
duction is switched on or off depending on the expression
of other genes， i．e． presence or absence of other proteins in
suMcient concentration．
Recently， by extending the notion of HPN， Matsuno et
al． ［16］ introduced hybrid functional Petri net （HFPN） in or－
der to give more intuitive and natural modeling method for
biological pathways than these existing Peui nets． Further－
more， we have been developing a software “Genomic Ob－
ject Net” for modeling and simulating biological pathways
based on the notion of the HFPN． GON equips the GUI spe－
cially designed for biological pathway modeling．
With GON， we have modeled and simulated many bi－
ological pathways， including the gene switch mechanisms
of 1 phage ［15］， the gene regulation for circadian rhythm in
Drosophila ［16］， the signal transduction pathway for apop－
tosis induced by the protein Fas ［16］， the glycolytic pathway
in E． coli wnh the lac operon gene regulatory mechanism
［17］， and Notch－Delta signaling mechanism of Drosophila
［18］．
From the nature of Petri net in visualization based de－
scription method， Petri net is acceptable modeling method
of biological pathways even for researchers in biology who
are not familiar with mathematical descriptions and pro－
gramming． On the other hand， a biological pathway con－
sists of a variety of biological reactions． Although our pre－
vious papers above gave a number of HFPN models of bi－
ological pathways， correspondence between biological re－
actions and HFPN components were not described explic－
itly． This paper presents “HZFPN component based modeling
method；’ which will help these researchers to constmct their
intended biological pathways more easily． Moreover， since，
with this method， researchers in biology and researchers in
system engineering can share their knowledge， it is expected
that this method promotes collaboration between these re－
searchers in different fields in discovering new biological
hypothesis which can not be produced without a help of
computer simulation．
As an example for demonstrating the HFPN compo－
nent based modeling method， this papar uses a biological
mechanisms “cell cycle of fission yeast ［1］．” Tyson’s group
has constructed several ordinary differential equation mod－
els of fission yeast cell cycle ［4］， ［24］， ［25］， in which at most
twenty proteins participate． Although they canied out nu一
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2920
merical simulations and analyzed the propenies of the cell
cycle system， around 10 known proteins were left behind
in their models and simulations． ln contrast， we have con－
stmcted the fission yeast cell cycle model so that it contains
a11 proteins so far examined biologically． Since fission yeast
is the most examined living organism on cell cycle mecha－
nisms， it can be said that the HFPN model of this paper is
the 1argest cell cycle model among the existing cell cycle
models．
2．Modeling Biological Pathw町with Hybrid Func・ tional Petri Net
2．1 Hybrid Functional Petri Net： Extension of Hybrid
Petri Net for Modeling Biological Reactions
Petri net is a network consisting of place， transition， arc，
and token． A place can hold tokens as its content． A tran－
sition has arcs co血ng from places and arcs going out from
the transition to some places． A transition with these arcs
defines a firing rule in terms of the contents of the places
where the arcs are attached．
Hybrid Petri net （HPN） ［2］ has two kinds of places dis－
crete place and continuous place and two kinds of transi－
tions， discrete transition and continuous transition． A dis－
crete place and a discrete transition are the same notions as
used in the traditional discrete Petri net ［30］． A continu－
ous place can hold a nonnegative real number as its con－
tent． A continuous transition fires continuously at the speed
of a parameter assigned at the continuous transition． The
graphical notations of a discrete transition， a discrete place，
a continuous transition， and a continuous place are shown in
Fig． 1， together with three types of arcs． A specific value
is assigned to each arc as a weight． When a normal arc
is attached to a discrete／continuous transition， w tokens are
transferred through the normal arc， in either of normal arcs
coming from places or going out to places． Note that， by
assigning different weights to these incoming and outgoing
arcs， different amounts can be fiowed in these two arcs． An
inhibitory arc with weight w enables the transition to fire
only if the content of the place at the source of the arc is
less than or equal to w． For example， an inhibitory arc can
be used to represent repressive activity in gene regulation．
A test arc does not consume any content of the place at the
source of the arc by firing． For example， a test arc can be
used to represent enzyme activity， since the enzyme itself is
discretetransition
Odiscrete
placecontinuoustransition
＠continuous place
一一一一ウ 一一一一ゆ◎ ・一・日…レ
normal arc inhibitory arc test arc
Fig． 1 Basic elements of HPN， HDN， and HFPN．
IEICE TRANS． FUNDAMENTALS， VOL．E87－A， NO．1 1 NOVEMBER 2004
not consumed．
Hybrid dynamic net（HDN）［5］has a similar structure
to the HPN， using the same kinds of places and transitions
as the HPN． The main di丘’erence betWeen HPN and HDN
is the firing of continuous transition． As described above，
fbr a continuous transition of HPN， the dif【’erent amounts of
tokens can be且owed through the two types of arcs， com－
ing fヒom！going out the continuous transition． In contrast，
the definition of HDN does not allow to transfbr different
amount through these two types of arcs． However， HDN has
the fbllowing firing fbature of continuous transitioll which
HPN does not have；‘‘the speed of continuous transition of
HDN can be given as a fUnction of values in the places．”
From血e above discussion it can be said that each of ウ
HPN and HDN has its own fbature fbr the且ring mechanism
of continuous transition． As a matter of fact both of these ヲ
fbatures of HPN and HDN are essentially required fbr mod－
eling common biological reactions． This motivated us to
propose hybrid fmctional Petri net（HFPN）［16］which in－
cludes both of these fbatures of HPN and HDN． Moreover シ
HFPN has the third feature fbr arcs， that is， a function of
values of the places can be assigned to any arc． This fbature
was originated from the idea in the paper［13］which was
introduced in order to realize the calculation of dynarnic bi－
ological cataly廿。 process on Petri net based biological path－
way modeling．
In fact， for any continuous transition in the biological
pathway model of且ssion yeast described in Section 3， same
amount of tokens flow in the incoming and outgoing arcs
to／from the transition． This means that this丘ssion yeast
model uses only feature of HDN． However， in the fbllowing，
we wiU use the notation‘‘HFPN”in order to keep consis－
tency with the biological pathway models so fa：r constmcted
［15］一［17］．
2．2 HFPN Components for B asic Biological Reactions
Many biological pathways can be constructed by mainly us－
ing the following six basic reactions： （1） protein composi－
tion and decomposition， （2） enzymic reaction， （3） protein
disintegration， （4） state switching， （5） state changing， and
（6） substance・ migration． HFPN components of these reac－
tions are listed in Fig． 2， and reaction speeds of these reac－
tions are summariZed血Table 1． The variables a，わ， c 1，
c2， d， e， and／are constants． Depending on the speed of
biological reactions to be modeled， the appropriate integer
in the range of 5 to 20 is assigned to each of six variables．
The variables V．．．， Km， and Ki are the maximum speeds， a
Michaelis constant， and a blocking constant of a Michaelis－
Menten equation ［1］， respectively． ln our model， we let
Vmax ＝ 2 and Km ＝ 1， and use a real number in the range of
O．4 to O．5 for the Ki．
In addition， a continuous transition with only one out－
put arc to a continuous place （e．g．， the continuous transitions
Ts attached to the continuous places Cigl， Cig2， Ruml，
Cdc18， and SCF－Popl／Pop2 in Fig．7） is used to model
constant protein production．
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FUJITA et al．： CELL CYCLE MODELING ON HYBRID FUNCTIONAL PETRI NET2921
prottm A protg／n BKiEl＞ 〈Ei＞
comp1ex AB
＠composition
＠
Vl
complexAB
decomposition
＠protein A
2
＠protein B
（a）Protein comptex formation
P筍A p㊨A
proteinA’ protdjnA’ （b）Enzyme reactions
prateindegradation
proteinA
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cited p’totein A
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nucleate A
＠
，
惜譜［＝コlellsvr
extranu’clearA
（DSubstrate migration
Fig．2 HFPN components for typical biological reactions． Contents of continuous places represent
concentrations of substances such as proteins， （b） Test arc is used for an enzymic reaction， since en－
zymes are not consumed by reactions． Two types of enzymic reactions “activating enzyme” and “block－
ing enzyme” are described． Fumhermore， activating enzyme reactions can be classified into two patterns
“constant concentration” and “variable concentrationJ’ Reaction speeds of them are given in Table 1．
（e） Discrete elements can be used for state changing reactions instead of constant elements． ln this case，
when a place gets a number of tokens enough to fire the transition， the reaction proceeds．
cycle changes from the G l phase to the S phase．
3． Modeling Cell Cycle of Fission Yeast with the HFPN
3．1 MPF Regulation
Figure 3 shows a whole HFPN model of the cell cycle reg－
ulation pathway of fission yeast． All initial values of places
and firing speeds of continuous transitions and delay times
of discrete transitions are tuned manually with repeating
simulations until concentration behaviors of proteins corre－
spond to the biological facts． Note that it is hard to decide
optimal values of these parameters， since data from biolog－
ical experiments are very insu伍cient to determine them．
Four blocks of the figure correspond to Fig．4， Fig．5，
Fig． 6， and Fig． 7． Small part at the central－right side of this
figure， which is not involved in any of these four blocks，
represents the phases of cell cycle． That is， when the
place G l has token（s）， the cell cycle is in G l phase． The
places S， G l， and G2 have similar meaning to the place
Gl． lf the places off－M and off－S have token（s）， the cell
cycle is not in the phases M and S， respectively． When
the place DNA－replication （DNA－replication－end） gets to－
ken（s）， DNA replication begins （ends）．
For expressing the reaction that the protein Cig 1 is pro－
duced only in the G l／S phase， the test arc is used from
the place G l to the transition representing the production
of the protein Cigl （refer to Fig．7）． lt is known that the
S phase does not begin unless the protein Cdc l 8 is broken
down ［22］，［23］． The test arc from the place representing
the breakdown of Cdc 18 to the transition DTis is used to
express this． Firing the transition DTis means that the cell
Figure 4 shows the HFPN model of MPF regulation mecha－
nism of budding yeast． A biological picture describing this
MPF regulation is found in the URL ［36］．
Protein complexformation （Fig． 2（a））
It is known that the proteins Cdc2 and Cdc 13 form a
complex ［26］． Complex formations can be modeled by con－
stmcting an HFPN model described in Fig． 2（a）． Based on
this， the complex formation is constmcted with the places
Cdc2， Cdc13， and Cdc2／Cdc13 at the left side in Fig．4．
T．hp reqqtion speed of this complex formation is given by1gCgg21；X2gglgldc2］＊5Cdci3］ with refening Table lt． The transitioris di， dS，
and d3 are used for representing degradation of the proteins
and the complex， which are attached to the places Cdc13，
Cdc2， and Cdcl 3／Cdc2， respectively． According to Table 1，
山・d・g・ad・t孟・n・peed・ar・gi・・n by th・f・・m・1・鼎， wh・・e
［P］ denotes the content of a place P at which the transition
f（）rdegradation is attached． The transitionsτ1 a皿d T3 are
used for productions of the proteins Cdc 13 and Cdc2， re－
spectively． The speeds of these transitions are set to 1．
The remaining part of Fig．4 were modeled by same
manner as described above． That is， a continuous place
of HFPN is placed for each substance， and places are con－
nected by transitions and arcs whose parameters （weights of
t［P］ represents the content of a place P．
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IEICE TRANS． FUNDmaNTALS， VOL．E87－A， NO．1 1 NOVEMBER 2004
2922
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FUJITA et al．： CELL CYCLE MODELING ON HYBRID FUNCTIONAL PETRI NET2923
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、⑳短評t1善 ；一．by－prctsgsome 十 I V28 i ： d i’圏r騨一一甲一一一一■．．■．開闘騨
⊂ut1ノ⊂ut2＋ubk四忙i i
signaLsl 1！AP（ノ⊂↓ 51p1／AP（ン⊂ 51pl
V29edi cutvcuaA－Jl r：Lid V31 rlS d
d
V30
Cutl一 XCut2
Td ［1］ ［1］dT
Fig． 6
蟹欝簿； Uii 1 ： l i st－ 1暖蕊謁詣説些㍑且＿．；
Srwl
響・d
港
T
d
HFPN model for M－phase progress．
一 DTio
lksl
L－s－
s－ss
stsL－
sss－st
l
I
l
l
SrwlXAPevC
V33 V34 d
signat＿Srwl！AP（ノ（
d
-
IEICE TRANS． FUNDAMENTALS， VOL．E87－A， NO．1 1 NOVEMBER 2004
2924
Oe2・3
Cd⊂13ノ（二d⊂2
Rumld［ト◎
福、
T［奄撃モрモ奄
→口d
リロ （＝d
1
τコ
π 由
Fig．3
T Cigl
d［トー
Cigl／Cdc2 d日←
季
8ve？，
撫Cig21CdcZ
口d
V37
Ruml d
V39SCF／Cdc18＋ubiqu’rtin
4÷脚
OOd
Cdc18 are broken
－by－proteasome
V41 1 V42
Cdc18＋P
。、伽躍τｸV44
SCFICdc18
ublquitin
XOd
lld
Ruml are broken
－by－proteasome
／Pop2
d
Fig．7 HF［PN model for MPF activating process and CKI effect．
Table 1 Transition speeds for HPN components of typical biological
reactions in Fig． 2．
Reaction 廿ansition speed
． ．モ盾高垂nSItlon 用1宰灘2ひ1＝ α
protein formation decomposition 胴302＝7
non－constant concentration 那4柳1503＝ わ
COnStant COnCentratiOn v泌ακ＊配403＝ κ渤＋鴻
㌦似蝦5 ， o?獅yymlC reaCtlOnS
competitive inhibi電ion ”4＝ v（1・繋）・・5
protein degradation 濯7り5＝7
protein state switching 1η8 彫9ひ6＝ττ07＝τ7
protein transformation 腕11ひ8＝7
substrate migration 加13ひ9＝一
Aー
arcs and speeds of transitions） are determined based on the
corresponding biological reactions． At transitions labeled
with the symbol T in Fig．4， production rates of the corre－
sponding proteins are assigned． ln the following， transitions
labeled with the symbols T or d have the same meaning as
above． The transitions vi， vi7， vis， and vig of Fig．4 are used’
for complex formations as shown in Fig． 2（a）．
Enzymic Reactions（Fig．2（b））
Transitionsひ2，03，05，ひ7，08，ひ9，ひ10，011，ひ20， v21，022，023，
ひ24，andひ60 are classified into the f6110wing three categories
according to the enzymic reaction types as shown in Table I．
（Non．constant concentration）Although proteins MPE CAK， and Cdc25 works as enzyme， activation level of
these proteins are not always constant． Based on this
fact血e formulas for non－constant concentration in Ta一 ラ
ble l are used fbr the transitions v8，09， vlo， and o60．
（Constant concentration）The formula for constant concen－
tration in Table l was used f6r the transitions v2，ひ3，
v5，07，011， v20，022， and o23， since concentrations of ki－
nases corresponding to these transitions keeps almost
constant leve1． The place Pyp3 represents the protein
Pyp3 which back up the protein Cdc25 ［21］．
（Competitive inhibition） For the transitions v2i and v24， the
formulas for competitive inhibition in Table 1 were
used from the following reasons． The transition v2i
represents the reaction that inhibits the protein Cdc25
from becoming non－active by the protein PP2A which
keeps almost constant concentration level throughout
the cell cycle ［14］． The transition v24 represents the
reaction that convert廿1e lamill into the lamina as山e
active MPF is broken down ［1］．
Protein degradation （Fig． 2（c））
At transitions labeled with the symbol d in Fig．4，
degradation rates of the corresponding proteins are assigned．
For the variable d， we should choose an integer not greater
than any other variables， since the speed of protein disinte－
gration should be slower than any other reactions． Accord－
ingly， 100 is used for the variable d in our model．
Protein state switching （Fig． 2（d））
State of some proteins will be switched between ac－
tive and non－active state depending on the concentration of
other protein． Each of the following combinations of transi－
tions （v4， vs）， （v21， v22）， （v23， v24）， and （v6， v7， vs） constitutes
the state switching shown in Fig． 2（d）． B asically， transition
speeds are assigned according to the reaction speeds given
in Table 1． However， if some enzymic reaction relates to
a state switching of protein， a formula for the enzymic re－
action will be used． For example， for the protein Weel， it
can be considered that both activation by the MPF and non－
activation by the Niml are enzymic reactions． Thus， formu－
las for enzymic reaction are used for the transitions v7 and
ひ8・
Protein transformation （Fig． 2（e））
The state of a protein will be transformed by reac－
tions such as phosphorylation and ubiquitination ［1］． For
example， a part of HFPN in Fig． 4 consisting of the places
-
FUJITA et al．： CELL CYCLE MODELING ON HYBRID FUNCTIONAL PETRI NET
Cdc13／Cdc2＋2P， Cdc131Cdc2＋3P， and active－CAK and
血e transition vg represents the reaction which transforms the
protein complex Cdc 13／Cdc2＋2P to Cdc 13／Cdc2＋3P by the
phosphorylation with the active CAK．
Substance migration （Fig． 2（b）
’IThis reaction is not included in Fig． 4， but used in the
model of checkpoint mechanism of Fig． 5． Refer to the next
subsection．
Usage ofdiscrete elements
The transitions vs and v7 are enzymic reactions with
the protein Num l and the protein complex active MPF． Al－
though it is known that each of these reactions does not oc－
cur wnhout a stimulation， the detai1 mechanism of the stim－
ulation is still unclear． ln such case， discrete elements are
used． At the top－left part of Fig．4， two discrete places and
two discrete transitions constitute a part which ca皿repre－
sents two states stimulate and not－stimulate． At each of the
transitions DTi and DT2， the time for state changing is as－
signed．
3．2 Checkpoint Mechanism
The concept of checkpoint was defined by Hartwell et al． in
1989 ［10］ as follows． “The events of the cell cycle of most
organisms are ordered into dependent pathways in which the
initiation of late events is dependent on the completion of
early events． Control mechanisms enforcing dependency in
the cell cycle are here called checkpoints 7’
Several checkpoint systems are known to work in the
cell cycle． Among them， we will focus on the system of
“SIM checkpoint” which coordinates the DNA synthesis and
the beginning of chromosome separation． The steps of S／M
checkpoint system are the following；
1．
2．
3．
the checkpoint mechanism is started by an initiator sig－
nal such as ultraviolet rays radiation，
the sensor senses an abnormal status in a cell triggered
by the initiator signal， and
the transducer mediates the abnormal status to the ef－
fector or血e targ・et．
In Fig． 5， two checkpoint mechanisms of fission yeast
“DNA damaging checkpoint” and “DNA replication check
point” are described together with the part of rvll？F activation
mechanism of Fig． 4． in the following， only the mechanism
for the DNA damaging checkpoint is explained， since these
two mechanisms have similar stmctures．
The initiator signal is generated at the beginning of S－
phase only when the DNA is damaged． This logic of mi－
tiator signal is realized with discrete elements at the top－left
corner of Fig． 5． The content of the discrete place p r rep－
resents the DNA damaging status （1：damaged， O：not dam－
aged）． Note that the formula S ＋1 is assigned at the in－
hibitory arc attached to the place p r． Since the content of
the discrete place is always set at 1， the place initiator－signal
can get token only when the discrete place S in Fig． 3 has at
2925
1east one token， where the place S indicates whether the cell
cycle is in S－phase （［S］＝1） or not （［S］＝O）．
When the concentration of the protein complexRad17SP， Rfc2， 一3， 一4， 一5 （place Rad l 7－Sp／Rfc2，3，4，5） ex－
ceeds some fixed level after detecting the initiator signal
（tra皿sition v46 and the place signal＿Rad17＿Sp／Rfc2，3，4，5），
the signal of DNA damaging is passed to the protein com－
plex Radl／Hus l／RadgSP （the places RadlIHusl／Rad9－Sp
and signal－Radl／Husll Rad9－Sp， and the transition v47）・
By transmitting this signal to the protein kinase Rad3 （the
place Rad3）， this kinase is activated血rough the phosphory－
lation （the place active－Rad3＋P）． The activated Rad3 phos－
phorylates the protein Chk l （the place C h k l）， activating
this protein （the place activenyChkl＋P）． Note that the pro－
tein complex Rad lIHus lIRadgSP and the proteins Rad3 and
Chk l are transducers．
The effector in this pathway is the protein Cdc25，
which is inactivated by the activated Chk l （the place
active－C h kl＋P） through the phosphorylation （the transi－
tion o52）． The inactivated Cdc25（1血e place Cdc25＋P）
is captured by the mobilizing factor Rad24SP （the place
Rad24－Sp． Since the captured Cdc25 （the placeCdc25／Rad24－S p） is transported to the outside of nucleus，
separating from the MPFt， it lose the ability to dephospho－
rize the Tyr 15 site． Eventually， the cell cycle is stopped at
G2－phase． By watching the discrete place DNA－damage？
（G2－STO P？）， we can know the status of the signal propaga－
tion of DNA damaging （the status of cell cycle temination
at G2－phase）．
3．3 HFPN Models for the M－Phase Progress and the CKI
Effect on MPF
Chromosome segregation and cell division occur in M－
phase in the order of dramatic events， prometaphase，
metaphase， and anaphase． On the other hand， CDK lnhibitor
（CKI） is a protein which works as a brake for the MPF， pre－
venting the MPF being out of control from the cell cycle reg－
ulation． Ruml is one of CKIs in yeast cell cycle， which is
started to produce at the anaphase of cell cycle， being com－
pletely degraded in the S－phase． That is， accumulation of the
Rum 1 suppresses血e MPF ac廿vadon du血g山e G l－phase．
Although both are the important processes in the cell
cycle， we only present the HFPN models as shown in Fig． 6
and Fig．7 due to the limitation of the space for this paper．
Refer to the webpage ［36］ for the details of them．
4． Simulations by Genomic Object Net
After describing an HFPN of the biological pathway to be
modeled， parameters of transition speed／delay and initial
values of places have to be detemined based on the biolog－
ical knowledge and／or the facts described in biological liter－
ature． ln general， many trial and error processes are required
tThis transportation corresponds to the reaction of “substance
migration” of Fig． 2（f）．
-
2926
1／g
l’s’1，
｝二l o
L．一4，．
Aotive－Cdc25
200 400time
600 800 iooo@1
（a）
．．s
80鱒一欄幽膳 8■otiv． MPF 一
・凹圏噂回・．馳鯛・．・． Cdo13／Cdo2＋3P＝MAP 一 Cdcで370 一
60 一舌軍
50 一召儒
40 一 騨一 一 一
8 評＝8 30Q0P0
一F”．κ＝！！i
m豊 填
@！iﾚ渦
喉 瀞／
@／
01
0 200 400 600 800 1000tim6
（b）
Fig．8 Abnormal behaviors of DNA－damaged fission yeast．
unti1 appropriate parameters for simulation are detemined．
Since GON provides the GUI specially designed for biologi－
cal pathway modeling， we can perform these processes very
easily and smoothly．
The constmcted HFPN model of fission yeast cell cycle
was simulated by using GON． With showing the simulation
result of a known behavior of DNA－damaged fission yeast，
the appropriateness of the constmcted HlrpN model is ver－
ified． ln addition， we will present a new biological hypoth－
esis obtained from the simulation with changing expression
level of the protein Rum l．
4．1 Verification of the Constructed Model： Observing B e－
haviors of DNA－Damaged Fission Yeast
Figure 8 shows concentration behaviors of the prote血s and
protein complexes related to MPF activation， where DNA－
damage occurs during a cell cyclet．
That is， as observed in Fig．8（a）， the amount of ac－
tive”Cdc25 is reduced rapidly due to the DNA－damage
which was realized by putting a token in the place p r in
Fig．5 at the time around 550． The signal triggered by mis
action propagates to the transition vs2 which reduces the
amount of the place active－Cdc25． The mechanism of this
signal propagation was explained in Sect． 3．2 with Fig． 5．
Figure 8（b） shows the behaviors of the places Cdc13，
active－M P F and Cdcl 3／Cdc2＋3P in Fig． 4． With compar－
ing Fig． 8（a）， we can observe that both the protein Cdc 13 and
the MPF stop oscillating at the time of the DNA－damage．
This observation reflects the biological fact that activation of
the protein complex Cdc 13／Cdc2 is controlled by the Cdc25
activation ［1］． This simulation result supports the appropri－
ateness of the constmcted HFPN model．
IEICE TRANS． FUNDAMENTALS， VOL．E87－A， NO．1 1 NOVEMBER 2004
50
一畠otiv●MPF 騨
Odc13／Cd62＋3P＝MAP r縣 ．一 Cdc13
40 一
‘ヨξ§o30
Q0
〆等
@ノ@’
f
4@’ …
畦 嘆 ’～ i夢 i
，葦
@’ 乏m を
m 蓬
パ
m1
10 需∠：コ 韮
?m1∫ 霧
mメ、
≠ 1
` ，
` 1m 【．．
ぎ 蓬
J ぎ 睾
xメ，0
0 200 400 600 800 1000timo
（a） wild type
50
一■otivo MPF 一
Cdo13／Cdc2中3P＝MAP 一 … Cdc1340 一
＝逼8＝30
Q0
一 〆
Q 〆～ メ
養
I1
警 渉タ茎 だ 垂〆i
8 ／ ～ 奏 ～婁 1 ’ 1 ’
10 一ム 潔 滅舛 4 魍0
0 200 400 600 800 1000timo
（b） deactivated
50
一量ctivo MPF Cdo1 一
3／Cdo2＋3P＝MAP Cdc1340 望
信8甥琶
30 一 ／i@ ／ i
浦窪 バ篠 ／
^ i
〆1／〆 i
O騙8
20
P0
一 ／ i
@ ノ 雲@ … ～L幽
ノ 1
` i
k凶メ i
@ …
@ ．r→0
0 200 400 600 800 毛 000tirno
（c） over expressed weakly
50一●ctivo＿MPF ・一…一Cdo13／CdG2＋3P＝MAP Cdc13
5＝ 40
一 ■暫ρ．・ 岬．． 哩壷”． 「冒需－「げ ，’い司 ’，い． ハh
o A一噛eゆ ．「晒一 一 嚇し ．一 ．@ゆ’Dか
」旨30 営 隠【8艦＝ 20 一
oo ’10 一
00 200 400 60◎ 800 1 000
tirne
（d） over expressed strongly
Fig．9 Simulation results of Rum l mutants．
4．2 Simulations of Mutants： Producing Hypotheses for
Further B iological Experiments
Figure 9 shows the simulation results of Rum l mutants． The
simulation result of （b） is obtained by removing the arc from
血e transition T4 to止e place Ruml in Fig．7． R血hermore，
by adding the transition with the arc going into the place
Ruml， the simulation results of （c） and （d） are obtained．
The values 2．0 and 1．0 are assigned to the transition for the
case of weak over expression （c） and strong over expression
（d）， respectively． B ehaviors of the protein Rum l for the fig－
ures （a）一（d） are presented in the website ［36］．
thFor each of Figs． 8（a） and （b）， the first waveform has a dif－
ferent shape compared to血e succeeding waveforms． Du血g血eperiod of the first waveform， values in the places converge to the
values with which the succeeding waveforms repeat periodically．
That is， the first waveform has to be neglected． The same situa－
tions occur in Figs． 9（a）， （b）， and （c）．
-
FUJITA et al．： CELL CYCLE MODELIING ON HYBRID FUNCTIONAL PETRI NIET
It is known from the biological experiments that cell
cycle does not stop even when the gene rum l is deactivated
［3］． The simulation result of （b） supports this fact． ln addi－
tion， it can be observed that the period of cell cycle of （b） is
shorter than that of （a） of wild type． This observation sup－
ports the fact that the protein Rum l controls the period of
cell cycle by breaking down the complex Cdc2／Cdc 13 ［19］．
Biological experiments in ［11］ elucidate the phe－
nomenon that over－expressing of the protein Rum l causes
the arrest of the MPF activation． The simulation result of
（d） supports this biological phenomenon． On the other had，
the period of simulation result of （c） is longer than that of
（d）， suggesting that the period of cell cycle becomes longer
if the over－expressing rate is decreased to some level． Note
that this suggestion has not been confirmed by any biologi－
cal experiment yet．
The above argument highlights that， with the help of
computer simulations， biologists can effectively produce hy－
potheses which will guide them in perfoming the further
biological experiments．
5． Conclusions
Researches on Petri net has a long history of nearly 40 years
from the paper by Dr． Petri ［27］ and it is mathematically
well－founded and practically well－established． B ased on the
researches on Petri nets， many tools have been developed by
researchers in concurrent technology ［37］．
GON is a biosimulation tool developed with inherit－
ing the tradition of the researches on Petri nets． These Petri
net tools so far developed generally have user－friendly GUIs
which allows us to describe complex concurrent systems
very easily and smoothly． GON inherits this feature of Peui
net， enabling to describe and manipulate biological pathway
naturally even for biologists who are not familiar with math－
ematical description and programming language． in con－
trast， E－Cell ［33］ and Gepasi ［20］， which are well－known
biosimulation systems， employ ordinary differential equa－
tions as description method for biological pathways． in ad－
dition， each of these systems does not equip the visual editor
as GON to describe biological pathways．
In this paper， we explained how biological pathways
can be described by HZFPN with the example of tl te cell cycle
of fission yeast． The constructed cell cycle pathway model
was simulated by GON． The simulation results suggest that
GON has very high potentiai to accelerate efficiencies of bi－
ological experiments drastically．
Acknowledgments
The authors would 1ike to thank Dr． Masao Nagasaki and
Mr． Atsushi Doi who gave us very usefu1 and suggestive
comments for simulations． This work is partially supported
by the Grand－in－Aid for Scientific Research on Priority Ar－
eas “Genome lnformation Science” from the Ministry of Ed－
ucation， Culture， Sports， Science and Technology in Japan．
2927
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2928
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IEICE TRANS． FUNDAMENTALS， VOL．E87－A， NO．1 1 NOVEMBER 2004
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享
Sachie Fujita received the B．S． and M．S．
degrees in lnformation Science from Yamaguchi
University in 2002 and 2004， respectively． She
joined Aso lnformation System Co． Ltd． While
a student， she had studied a method for biologi－
cal pathway modeling and simulation with Petri
nets．
Mika Matsui received the B．E． and M．E． de－
grees in Electrical Engineering from Toyohashi
University of Technology in 1993 and 1995， re－
spectively． ln 1995， she joined Omron Corpora－
tion． From 1998 to 2004， she was a research as－
sociate of Oshima National College of Maritime
Technology． She is currently working towards
a Ph．D degree at Graduate School of Science
and Engineering， Yarnaguchi University． She is
a member of Japanese Society for Bioinformat－
ics．
Hireshi Matsuno received the B．E and
M．E． degrees in Electronics from Yarnaguchi
University in 1982 and 1984， respectively． He
received the Ph．D degree from Kyushu Univer－
sity in 1994． From 1984 to 1995， he worked
at Yamaguchi Junior College and Oshima Na－
tional College ofMaritime Technology． ln 1995，
Dr． Matsuno joined Faculty of Science， Yama－
guchi University， and he is currently an associate
professor． His current interests includes systems
biology and wireless LAN communications． He
is a member of IEEE， IPSJ， and Japanese Society for Bioinformatics．
Satoru Miyano received the B．S． degree in
1977， M．S． degrees in 1979， and Dr． Sci． in 1984
all in Mathematics from Kyushu University． He
is presently a Professor of Human Genome Cen－
ter， institute of Medical Science， The Univer－
sity of Tokyo． He received IBM Science Award
and Sakai Special Award both in 1994． His re－
search areas are Bioinformatics， Computational
Biology， Genome lnformatics， and Discovery
Science． He is a member of ACM， IPSJ， and
Japanese Society for Bioinformatics．
