A Mul iscale Modeling F amewo k
Based on P Sys ems
F ancisco Jos´e Rome o-Campe o1, Jamie Twyc oss1,2,
Hongqing Cao1, Jona han Blakes1, and Na alio K asnogo 1
1Au oma ed Scheduling, Op imisa ion and Planning Resea ch G oup
School o Compu e Science, Jubilee Campus, Uni e si y o No ingham
No ingham NG8 1BB, Uni ed Kingdom
2Cen e o Plan In eg a i e Biology
Su on Boning on Campus, Uni e si y o No ingham
No ingham LE12 5RD, Uni ed Kingdom
{ xc,jp ,hxc,j b,nxk}@cs.no .ac.uk
Abs ac . Cellula sys ems p esen a highly complex o ganiza ion a
diffe en scales including he molecula , cellula and colony le els. The
complexi y a each one o hese le els is igh ly in e ela ed. In eg a i e
sys ems biology aims o ob ain a deepe unde s anding o cellula sys ems
by ocusing on he sys emic and sys ema ic in eg a ion o he diffe en
le els o o ganiza ion in cellula sys ems.
The diffe en app oaches in cellula modeling wi hin sys ems biology
ha e been classified in o ma hema ical and compu a ional amewo ks.
Specifically, he me hodology o de elop compu a ional models has been
ecen ly called execu able biology since i p oduces execu able algo i hms
whose compu a ions esemble he e olu ion o cellula sys ems.
In his wo k we p esen P sys ems as a mul iscale modeling ame-
wo k wi hin execu able biology. P sys em models explici ly speci y he
molecula , cellula and colony le els in cellula sys ems in a ele an and
unde s andable manne . Molecula species and hei s uc u e a e ep-
esen ed by objec s o s ings, compa men aliza ion is desc ibed using
memb ane s uc u es and finally cellula colonies and issues a e modeled
as a collec ion o in e ac ing indi idual P sys ems.
The in e ac ions be ween he componen s o cellula sys ems a e de-
sc ibed using ew i ing ules. These ules can in u n be g ouped oge he
in o modules o cha ac e ize specific cellula p ocesses. One o ou cu -
en esea ch lines ocuses on he design o cell sys ems biology models
exhibi ing a p efixed beha io h ough he au oma ic assembly o hese
cellula modules. Ou app oach is equally applicable o syn he ic as well
as sys ems biology.
1 In oduc ion
Models in sys ems biology has been ecen ly classified acco ding o hei se-
man ics in o deno a ional and ope a ional models [6]. Models wi h deno a ional
seman ics a e he classical app oach in modeling cellula sys ems which uses a
se o equa ions o desc ibe how he quan i ies o he diffe en molecula species
a e ela ed o each o he o e ime. The classical example a e o dina y and pa -
ial diffe en ial equa ions. In his case he beha io o he sys em is ob ained by
app oxima ing nume ically hese equa ions. On he o he hand, compu a ional
models ha e ope a ional seman ics which desc ibe he beha io o he sys em
using an algo i hm o lis o ins uc ions ha can be execu ed by an abs ac
machine. The models de eloped wi hin his las amewo k has been e med
ecen ly execu able biology [6]. In his case a mo e de ailed desc ip ion o he
p ocesses p oducing he beha io o he sys em is p o ided.
Se e al o mal compu a ional app oaches ha e been p oposed o model cellu-
la sys ems like Pe i ne s [10] and p ocess algeb a [19]. They mainly ocus on
sys em specifica ion a he molecula le el: memb anes, compa men aliza ion
and cellula colonies a e seldom desc ibed. This ac makes i difficul o s udy
mul icellula sys ems whose unc ion is de e mined by molecula in e ac ions.
Memb ane compu ing is a b anch o na u al compu ing inspi ed di ec ly om
he s uc u e and unc ioning o he li ing cell [14]. I has been applied o cellula
modeling as one o he ew compu a ional amewo ks which p esen s an in eg a-
i e app oach o mul iscale sys ems anging om he molecula o he mul icellu-
la le el. Specifically, i ep esen s he molecula in e ac ion le el o li ing cells
using objec s o s ings and ew i ing ules; he compa men al/cellula le el
using memb anes; and he colony le el using collec ions o memb anes called
memb ane s uc u es. The de ices o his compu a ional pa adigm a e e e ed
o as Psys ems. Al hough mos esea ch in P sys ems ocuses on he s udy o he
compu a ional powe o he diffe en p oposed a ian s, ecen ly hei applica ion
as a modeling o malism o cellula sys ems is eme ging [3,4,7,11,17,20,21,15].
In his pape we discuss h ough a unning example he use o P sys ems as a
mul iscale modeling amewo k o cell sys ems biology models.
The pape is o ganized as ollows. S ochas ic P sys ems o cellula model-
ing a e in oduced in Sec ion 2. Sec ion 3 p esen s he unning example used
h oughou his pape . The modeling p inciples in P sys ems a e desc ibed in
Sec ion 4. Modula iza ion in P sys ems is b iefly discussed in Sec ion 5. Finally,
conclusions and u u e wo k a e discussed in Sec ion 6.
2 S ochas ic P Sys ems
The o iginal s a egy o he applica ion o he ew i ing ules in P sys ems was
based on maximal pa allelism and non-de e minism [13]. This s a egy does no
ep esen he a e a which molecula in e ac ions ake place as e e y objec ha
can e ol e acco ding o any ule mus e ol e in a single compu a ion s ep, wi h-
ou aking in o accoun ha some molecula in e ac ions a e mo e equen han
o he s. Mo eo e , he eal ime e olu ion o cellula sys ems is no cap u ed as
all he compu a ion s eps a e assumed o be o he same ime leng h, neglec ing
he ac ha some molecula in e ac ions a e as e han o he s.
Diffe en s a egies o he applica ion o he ew i ing ules in P sys ems ha e
been s udied [5,8]. Specifically, a sequen ial s ochas ic s a egy based on Gille-
spie’s heo y o s ochas ic kine ics [9] was in oduced in o de o o e come he wo
p e ious p oblems when de eloping a modeling amewo k o cellula sys ems bi-
ology based on P sys ems [16]. He e we e e o his a ian as s ochas ic P sys ems.
Defini ion 1 (S ochas ic P Sys ems). A S ochas ic P sys em is a cons uc :
Π=((Σobj,Σ
s ),L,μ,M
l1,...,M
lm,(Robj
l1,R
s
l1),...,(Robj
lm,R
s
lm)),
whe e:
•Σobj is a fini e alphabe o objec s ep esen ing molecula species whose in-
e nal s uc u e is no ele an in he unc ioning o he sys em unde s udy.
•Σs is a fini e alphabe o objec s ep esen ing ele an pa s o some molec-
ula species in he sys em. These objec s a e a anged in o s ings desc ibing
he s uc u e o molecula species.
•L={l1,...,l
m}is a fini e alphabe o symbols ep esen ing compa men
labels used o iden i y compa men classes. Compa men s wi h he same
label sha e he same class, i.e., se o ew i ing ules and ini ial mul ise s.
•μis a memb ane s uc u e consis ing o n≥1memb anes defining compa -
men s iden ified in a one o one manne wi h alues om {1,...,n}and
labeled wi h elemen s om L.
•Ml =(w ,s
), o each1≤ ≤m, is he ini ial s a e o he compa men s
om he class iden ified by label l ,whe ew ∈Σ∗
obj is a fini e mul ise o
indi idual objec s and s is a fini e se o s ings o e Σs .Amul ise o
objec s, obj is ep esen ed as obj =o1+o2+...+opwi h o1,...,o
p∈Σobj.
S ings a e ep esen ed as ollows s1·s2···sqwhe e s1,...,s
q∈Σs .
•Robj
l ={ obj,l
1,...,
obj,l
kobj,l }, o each1≤ ≤m, is a fini e mul ise o
ew i ing ules on mul ise s o objec s associa ed wi h compa men s o he
ype specified by he label l . The ew i ing ules on mul ise s o objec s a e
o he ollowing o m:
obj,l
j:obj1[obj2]l
cobj,l
j
−→ obj
1[obj
2]l(1)
wi h obj1,obj
2,obj
1,obj
2some fini e mul ise s o objec s om Σobj and lala-
bel om L. These ules a e mul ise ew i ing ules ha ope a e on bo h sides
o memb anes, ha is, a mul ise obj1placed ou side a memb ane labeled by
landamul ise obj2placed inside he same memb ane can be simul aneously
eplaced wi h a mul ise obj
1andamul ise obj
2, espec i ely.
No e ha a cons an cobj,l
jis associa ed specifically wi h each ule. This
cons an will be e e ed o as s ochas ic cons an and is key o p o ide P
sys ems wi h a s ochas ic ex ension as i will be used o compu e he p ob-
abili y and ime needed o apply each ule. This cons an depends only on
he physical p ope ies o he molecules and compa men s in ol ed in he
eac ion desc ibed by he ule like empe a u e, p essu e, pH, olume, e c.
•Rs
l ={ s ,l
1,...,
s ,l
ks ,l }, o each1≤ ≤m, is a fini e se o ew i ing
ules on mul ise s o s ings and objec s associa ed wi h compa men s o he
ype defined by l and o he ollowing o m:
s ,l
j:[obj +s ]l
cs ,l
j
−→ [obj+s ;s 
1+...+s 
s]l(2)
wi h obj, objmul ise s o objec s o e Σobj and s , s ,s 
1,...,s

ss ings
o e Σs . These ules ope a e on bo h mul ise s o objec s and s ings. The
objec s obj a e eplaced by he objec s obj. Simul aneously a subs ing s is
eplaced by s whe eas he s ings s 
1,...,s

sa e p oduced o o m pa
o he con en o he compa men . In he same way as o ew i ing ules on
mul ise s o objec s a s ochas ic cons an cs ,l
jis associa ed wi h each ule.
The p e ious defini ion is p o ided wi h a s ochas ic s a egy o he applica ion
o he ew i ing ules by ex ending he Gillespie algo i hm o he mul icom-
pa men al s uc u e o P sys ems. The esul ing algo i hm has been e e ed o
as he Mul icompa men al Gillespie Algo i hm (MGA) [16]. The Gillespie algo-
i hm [9] can only be applied di ec ly in a single, fixed and well mixed olume.
In ou app oach he fi s s ep consis s o ea ing each compa men defined by
a memb ane as a fixed and well mixed olume whe e he ew i ing ule o be ap-
plied and he elapsed ime be o e i s applica ion is compu ed using he Gillespie
Di ec Me hod. Ou algo i hm hen applies he co esponding ules ollowing he
o de de e mined by hese wai ing imes. A e he applica ion o each ule he
algo i hm ecompu es he ules o be applied and he wai ing imes in he com-
pa men s affec ed by he applica ion o he las ule using he Gillespie Di ec
Me hod. Finally, he MGA hal s when a p efixed simula ion ime is eached o
no u he ules can be applied.
3 Running Example
In o de o illus a e ou modeling amewo k we will use an abs ac gene eg-
ula ion sys em inspi ed om he unc ioning and s uc u e o he lac ope on in
Esche ichia coli (E. coli). This ope on consis s o h ee s uc u al genes, lacZ,
lacY and lacA, loca ed sequen ially on he genome and ansc ibed in o one
single mRNA. Thei p o ein p oduc s a e in ol ed in he sensing, up ake and
me abolism o lac ose. The ansc ip ion o he lac ope on is bo h posi i ely and
nega i ely egula ed and i is conside ed a canonical example o gene ansc ip-
ion egula ion in p oka yo es [18].
The linea s uc u e o he lac ope on (Figu e 1) s a s wi h a egion called cap
whe e he ac i a o p o ein CRP binds and inc eases he a e o ansc ip ion.
Following his si e he e is an ope a o sequence ha we will e e o as op whe e
he ep esso p o ein LacI binds o s op ansc ip ion. The s uc u al genes lacZ,
lacY and lacA hen ollow. The fi s gene lacZ codifies he enzyme β-galac osidase
in ol ed in he me abolism o lac ose by clea ing i in o glucose and galac ose;
allolac ose appea s as a byp oduc o his eac ion. The p o ein p oduc o he
second gene lacY is a pe mease ha associa es o he cell memb ane and ac s
as a pump anspo ing lac ose in o he cell. The unc ion o he p o ein coded
in he las gene lacA is no ye ully unde s ood.
The egula ion o he lac ope on allows E. coli o exp ess he genes in he
ope on only when i is mo e beneficial o he cell. In he absence o lac ose in
he media he ep esso LacI binds o he ope a o op p e en ing he s uc u al
⇓
cap.op.lacZ.lacY.lacA
Fig. 1. A schema ic ep esen a ion o he s uc u e o he lac ose ope on ( op) and i s
ep esen a ion as a s ing (bo om)
genes om being ansc ibed since hey a e no needed unde hese condi ions.
Ne e heless, occasionally he ep esso d ops om he ope a o p oducing a
basal ansc ip ion o he ope on.
When lac ose becomes a ailable i s a s o be anspo ed inside he cell by
he basal numbe o LacY p o eins on he cell su ace. Once in he cy oplasm
i in e ac s wi h he basal numbe o β-galac osidase p oducing as a byp oduc
allolac ose. Allolac ose in u n binds o he ep esso LacI and changes i s s a e
so i canno bind o he ope a o allowing ansc ip ion o he s uc u al genes.
The esul ing inc ease in p oduc ion o LacY and β-galac osidase o ms a posi i e
eed-back loop inc easing he numbe o allolac ose molecules which in e ac wi h
he ep esso s p e en ing p ema u e e mina ion o ansc ip ion.
The lac ope on is also unde posi i e egula ion by he p o ein CRP. This
p o ein is ac i a ed by he glucose anspo sys em and when ac i e i binds o
he cap si e acili a ing ansc ip ion. E en in he p esence o lac ose i glucose
is p esen in he media CRP will no be ac i e as he anspo sys em will
be occupied, pumping glucose in o he cell. The e o e CRP will no bind o
he ope on o assis ansc ip ion. Only in he p esence o lac ose and absence
o glucose will CRP be ac i e and bound o he ope on, p oducing he ull
ansc ip ion o he ope on.
This gene egula ion sys em will be used in he ollowing sec ion as he unning
example illus a ing ou modeling p inciples.
4 Modeling P inciples
The complexi y o cellula sys ems is o ganized in o diffe en le els anging om
he molecula o he cellula and colony scales. These le els o complexi y a e no
independen ins ead hey a e igh ly in e ela ed influencing each o he di ec ly.
In his espec , s ochas ic P sys ems p esen an in eg a ing mul iscale modeling
amewo k which explici ly specifies he molecula , cellula and colony le els in
cellula sys ems in a ele an and unde s andable manne .
One o ou esea ch lines consis s o he de elopmen o in eg a i e modeling
p inciples wi hin he modeling amewo k o s ochas ic P sys ems. Mo e specifi-
cally we will p esen some ideas on how o desc ibe molecula species, cellula e-
gions and compa men s, molecula in e ac ions, gene exp ession con ol and cell
colonies. Ou unning example will be used o illus a e ou modeling p inciples.
•Molecula species: These a e specified as indi idual objec s o s ings o
objec s. Molecules wi h an in e nal s uc u e ha is ele an in he

unc ioning o he sys em a e specified using s ings. Fo example, gene
ope ons wi h a linea s uc u e consis ing o p omo e s, ope a o s, an-
sc ip ion/ ansla ion s a ing poin s, e c, o he wise molecula species a e
desc ibed using indi idual objec s.
Table 1. Specifica ion o he molecula species in he lac ope on
Molecula Species Objec
RNA Polyme ase RNAP
Ribosome Rib
Rep esso LacI
Ac i a o CRP CRP∗
LacZ p oduc LacZ
LacY p oduc LacY
LacA p oduc LacA
Lac ose Lac
Allolac ose Allolac
Glucose Gluc
Glucose anspo
sys em Gluc
Complex glucose
anspo sys em Gluc-GT S
Complex lac ose
LacY p oduc Lac -LacY
Complex lac ose
LacZ p oduc Lac -LacZ
Complex lac ose
LacZ p oduc Lac -LacZ
Complex allolac ose
ep esso Allolac-LacI
Ope on si e Objec
Ac i a o binding si e cap
Occupied ac i a o
binding si e capCRP∗
Rep esso binding si e op
Occupied ep esso
binding si e opLacI
lacZ gene lacZ
lacY gene lacY
lacA gene lacA
lacZ mRNA mlacZ
lacY mRNA mlacY
lacA mRNA mlacA
Running example: The diffe en molecula species in ou example will be
specified acco ding o his modeling p inciple. On he one hand, he p o eins
and complexes o p o eins in ol ed in he egula ion and exp ession o he lac
ope on a e specified as indi idual objec s since we a e no in e es ed in hei
in e nal s uc u e (Table 1). On he o he hand, each componen o he lac
ope on will be desc ibed using an objec such ha he lac ope on s uc u e
is specified as a s ing con aining hese objec s in he specific o de hey can
be ound in E. coli’s genome (Figu e 1).
•Memb anes: Compa men aliza ion and memb anes a e undamen al in
he s uc u al o ganiza ion and unc ioning o li ing cells. Memb anes do no
ac as passi e bounda ies o cells and compa men s; ins ead hey play a key
ole in he egula ion o he me abolism and in o ma ion p ocessing be ween
he ou side and he inside o compa men s. P sys ems cons i u es one o
he ew compu a ional amewo ks which explici ly specifies compa men s
and memb anes. Fo ins ance, P sys ems ha e been used o s udy selec i e
up ake o molecules om he en i onmen [20], signalling a he cell su ace
[12] and colonies o in e ac ing bac e ia which communica e by sending and
ecei ing diffusing signals [2,21]. In gene al P sys em memb anes a e used o
define ele an egions in cellula sys ems and he e o e hey do no always
co espond o eal cell memb anes al hough no mally hey do.
Running example: In he lac ope on gene egula ion sys em he e a e wo
ele an egions. Namely, he bac e ium su ace whe e LacY and GT S ac
as pumps anspo ing lac ose and glucose in o he cell and he aqueous
in e io o cy oplasm whe e he ope on is loca ed oge he wi h he diffe en
ansc ip ion ac o s and p o eins. These wo egions a e ep esen ed using
wo memb anes embedded one inside he o he o desc ibed he s uc u e o
an E. coli bac e ium (Figu e 2).
Fig. 2. G aphical ep esen a ion o he memb ane s uc u e speci ying an E. coli
bac e ium
•Molecula p ocesses consis ing o p o ein-p o ein in e ac ions and
p o ein ansloca ion: Such p ocesses a e no mally desc ibed in P sys ems
using ew i ing ules on mul ise s o objec s. Ou P sys em modeling ame-
wo k aims a p o iding a comp ehensi e and ele an ule-based schema o
he mos common molecula in e ac ions aking place in li ing cells. Mo e
specifically, ou app oach ocuses on he ans o ma ion and deg ada ion o
molecula species, he o ma ion and dissocia ion o complexes, and he basic
p ocesses o communica ion and anspo be ween diffe en compa men s
in cellula sys ems (Table 2).
Running example: The p o ein-p o ein in e ac ions in ou gene egula ion
sys em a e desc ibed using he ew i ing ules on mul ise s o objec s p e-
sen ed in Table 3. Rules 29,
30,
31 and 32 a e examples o complex o ma-
ion and dissocia ion ules. The deg ada ion and dilu ion o diffe en p o eins
Table 2. P sys em ule-based schemas o he mos common molecula in e ac ions
Molecula In e ac ion PSys emRules
T ans o ma ion and Deg ada ion [a]l
c
−→ [b]l[a]l
c
−→ []
l
Complex o ma ion and dissocia ion [a+b]l
c
−→ [c]l[c]l
cd
−→ [a+b]l
Diffusion in and ou a[]
l
cin
−→ [a]l[a]l
cou
−→ a[]
l
Binding and debinding a[b]l
clb
−→ [c]l[c]l
cld
−→ a[b]l
Rec ui men and eleasing a[b]l
c
−→ c[]
lc[]
l
c l
−→ a[b]l
is specified in ules 22,
23 and 24. Finally, ac i e up ake o glucose and lac-
ose a e modeled using he binding and eleasing ules 27,
28,
33 and 34.
•Gene exp ession con ol: The sensing o signals and he p ocessing o
he in o ma ion hey con ey is pe o med in li ing cells h ough molecula
in e ac ions o he ype p esen ed in Table 2. The esponse o cells o hese
signals consis s o he exp ession o app op ia e p o eins codified in specific
genes. Gene exp ession con ol has been desc ibed in P sys ems using ei he
ew i ing ules on mul ise s o objec s o ew i ing ules on mul ise s o
objec s and s ings acco ding o he s uc u al o ganiza ion o he genes in
he sys em unde s udy. Tables 4 and 5 p esen s hese wo al e na i es o
he specifica ion o he mos impo an p ocesses in gene exp ession con ol;
ansc ip ion ac o binding and debinding, ansc ip ion and ansla ion.
F om a simplis ic poin o iew he p ocesses in ol ed in ansc ip ion ac-
o binding and debinding, ansc ip ion and ansla ion can be ep esen ed
by indi idual ew i ing ules on mul ise s o objec s (Table 4). Ne e heless,
hese p ocesses a e e y complex and hey consis o diffe en s ages like op-
e a o /p omo e ecogni ion by ansc ip ion ac o s and RNA polyme ase,
ansc ip ion/ ansla ion ini ia ion/ e mina ion, elonga ion, e c. A mo e ac-
cu a e and de ailed desc ip ion o all hese p ocesses is achie ed by using
ew i ing ules on mul ise s o s ings and objec s o he o m o he ules in
Table 5.
Running Example: The gene egula ion con ol in he lac ope on is modeled
using he ew i ing ules on mul ise s o objec s and s ings gi en in Table 3.
Mo e specifically, he binding and debinding o he ac i a o and ep esso
o hei co esponding binding si es is ep esen ed using ules 3,
4,
7and
8. T ansc ip ion ini ia ion in he p esence and absence o he p omo e si e
occupied by he ac i a o CRP∗is specified using ules 1,
2,
5and 6.
The ansc ip ion o he s uc u al genes lacZ,lacY and lacA is desc ibed
by he ules 9,
10,
11 and 12. Finally, ansla ion and mRNA deg ada ion
is modeled wi h he ules 13 - 21.
•Cell colonies: The las le el o o ganiza ion ha has been ep esen ed using
P sys ems consis s o cellula sys ems whe e cells o m colonies by in e ac ing
Table 3. Lac Ope on Regula ion Rules
N . Rule S ochas ic Cons an
1:[RNAP +cap]b
c1
−→ [cap.RNAP ]bc1=5×10−3min−1
2:[cap.RNAP ]b
c2
−→ [RNAP +cap]bc2=1min−1
3:[CRP∗+cap]b
c3
−→ [capCRP∗]bc3=16.6min−1
4:[capCRP∗]b
c4
−→ [CRP∗+cap]bc4=10min−1
5:[RNAP +capCRP∗]b
c5
−→ [capCRP∗.RNAP ]bc5=0.2min−1
6:[capCRP∗.RNAP ]b
c6
−→ [RNAP +capCRP∗]bc6=1min−1
7:[LacI +op]b
c7
−→ [opLacI ]bc7= 166min−1
8:[opLacI ]b
c8
−→ [LacI +op]bc8=0.1min−1
9:[RNAP.op]b
c9
−→ [op.RNAP ]bc9=3min−1
10 :[RNAP.lacZ]b
c10
−→ [lacZ.RNAP;mlacZ]bc10 =0.78min−1
11 :[RNAP.lacY ]b
c11
−→ [lacY.RNAP;mlacY ]bc11 =1.92min−1
12 :[RNAP.lacA]b
c12
−→ [RNAP +lacA;mlacA]bc12 =4min−1
13 :[Rib +mlacZ]b
c13
−→ [Rib.mlacZ]bc13 =0.12min−1
14 :[Rib +mlacY ]b
c14
−→ [Rib.mlacY ]bc14 =0.12min−1
15 :[Rib +mlacA]b
c15
−→ [Rib.mlacA]bc15 =0.12min−1
16 :[Rib.mlacZ]b
c16
−→ [Rib +LacZ +mlacZ]bc16 =0.12min−1
17 :[Rib.mlacY ]b
c17
−→ [Rib +LacY +mlacY ]bc17 =1.73min−1
18 :[Rib.mlacA]b
c18
−→ [Rib +LacA +mlacA]bc18 =3.55min−1
19 :[mlacZ]b
c19
−→ []
bc19 =6×10−3min−1
20 :[mlacY ]b
c20
−→ []
bc20 =6×10−3min−1
21 :[mlacA]b
c21
−→ []
bc21 =6×10−3min−1
22 :[LacZ ]b
c22
−→ []
bc22 =6.9×10−2min−1
23 :[LacY ]b
c23
−→ []
bc23 =6.9×10−2min−1
24 :[LacA ]b
c24
−→ []
bc24 =6.9×10−2min−1
25 :[LacY ]b
c25
−→ LacY []
bc25 =1min−1
26 :LacY []
b
c26
−→ [LacY ]bc26 =0.7min−1
27 :Lac [LacY ]s
c27
−→ [Lac -LacY ]sc27 =10min−1
28 :Lac -LacY []
b
c28
−→ LacY [Lac ]bc28 =10min−1
29 :[Lac +LacZ ]b
c29
−→ [Lac -LacZ ]bc29 =10min−1
30 :[Lac -LacZ ]b
c30
−→ [Allolac +LacZ ]bc30 =10min−1
31 :[Allolac +LacI ]b
c31
−→ [Allolac-LacI ]bc31 =1min−1
32 :[Allolac-LacI ]b
c32
−→ [Allolac +LacI ]bc32 =10
−4min−1
33 :Gluc [GT S ]s
c33
−→ [Gluc-GT S ]sc33 =1min−1
34 :Gluc-GT S []
b
c34
−→ GT S [Gluc ]bc34 =10min−1
35 :GT S [CRP ]b
c6
−→ GT S [CRP∗]bc35 =6.9×10−3min−1
36 :[CRP∗]b
c6
−→ []
bc36 =0.069min−1
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