Modelling gene exp ession con ol using P sys ems:
The Lac Ope on, a case s udy
F ancisco Jos´e Rome o-Campe o, Ma io J. P´e ez-Jim´enez ∗
Resea ch G oup on Na u al Compu ing, Depa men o Compu e Science and A ificial In elligence,
Uni e si y o Se illa, A da. Reina Me cedes s/n, 41012 Se illa, Spain
Abs ac
In his pape P sys ems a e used as a o mal amewo k o he speci ica ion and simula ion o biological sys ems. In pa icula ,
we will deal wi h gene egula ion sys ems consis ing o p o ein–p o ein and p o ein–DNA in e ac ions ha ake place in di e en
compa men s o he hie a chical s uc u e o he li ing cell o in di e en indi idual cells om a colony. We will explici ly model
ansc ip ion and ansla ion as concu en and disc e e p ocesses using ew i ing ules on mul ise s o objec s and s ings. Ou
app oach akes in o accoun he disc e e cha ac e o he componen s o he sys em, i s andom beha iou and he key ole played
by memb anes in p ocesses in ol ing signalling a he cell su ace and selec i e up ake o subs ances om he en i onmen . Ou
sys ems will e ol e acco ding o an ex ension o Gillespie’s algo i hm, called Mul icompa men al Gillespie’s Algo i hm. The well
known gene egula ion sys em in he Lac Ope on in Esche ichia coli will be modelled as a case s udy o benchma k ou app oach.
Keywo ds: P sys ems; Sys ems biology; Gillespie’s algo i hm; Gene exp ession con ol; Lac Ope on
1. In oduc ion
Memb ane Compu ing is an eme gen b anch o Na u al Compu ing in oduced by P˘
aun (2000). Since hen i has
ecei ed impo an a en ion om he scien i ic communi y. In ac , Memb ane Compu ing has been selec ed by he
Ins i u e o Scien i ic In o ma ion, USA, as a as Eme ging Resea ch F on in Compu e Science in Oc obe 2003.
This new model o compu a ion s a s om he assump ion ha he p ocesses aking place in he compa men al
s uc u e o a li ing cell can be in e p e ed as compu a ions. The de ices o his model a e called P sys ems. Roughly
speaking, a P sys em consis s o a cell-like memb ane s uc u e, in he compa men s o which one places mul ise s o
objec s which e ol e acco ding o gi en ules.
Mos a ian s o memb ane sys ems ha e been p o ed o be compu a ionally comple e, ha is equi alen in powe
o Tu ing machines, and compu a ionally e icien , ha is able o sol e compu a ionally ha d p oblems in polynomial
ime. Al hough mos esea ch in P sys ems concen a es on compu a ional powe s, la ely hey ha e been used o
model biological phenomena (Bianco e al., 2005; Ciobanu e al., 2006; Pescini e al., 2006; P´
e ez-Jim´
enez and
Rome o-Campe o, 2006).
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (F.J. Rome o-Campe o), [email p o ec ed] (M.J. P´
e ez-Jim´
enez).
As P sys ems a e inspi ed om he unc ioning o he li ing cell, i is na u al o conside hem as modelling ools
o biological sys ems, wi hin he amewo k o sys ems biology, being an al e na i e o mo e classical app oaches
such as o dina y di e en ial equa ions (ODEs). Di e en ial equa ions ha e been used success ully o model kine ics
o con en ional mac oscopic chemical eac ions. Ne e heless, he e is an implici assump ion o con inuously a ying
chemical concen a ion and de e minis ic dynamics. Two c i ical cha ac e is ics o his app oach a e ha he numbe
o molecules o each ype in he eac ion mix is la ge and ha o each ype o eac ion in he sys em, he numbe o
eac ions is la ge wi hin each obse a ion in e al, ha is eac ions a e as .
When he numbe o pa icles o he eac ing species is low and eac ions a e slow, which is equen ly he case in
gene exp ession con ol in bac e ia and i uses, bo h o he p e ious p esump ions a e in alid and he de e minis ic
con inuous app oach o chemical kine ics is ques ionable. Ins ead one has o ecognise ha he indi idual chemical
eac ion s eps occu disc e ely and a e sepa a ed by ime in e als o andom leng h.
Besides i is well known ha ansc ip ion and ansla ion in bac e ia a e concu en and disc e e p ocesses ha
p oduce a delay in he exp ession o genes. The o de in which di e en genes a e ound in ope ons on he genome o
Esche ichia coli is also a ele an ea u e in he unc ioning o he gene exp ession con ol. All hese cha ac e is ics
a e no easily speci ied and simula ed in classical app oaches like ODEs.
In con as o di e en ial equa ions, P sys ems a e an uncon en ional model o compu a ion which akes in o
conside a ion he disc e e cha ac e o he quan i y o componen s o he sys em by using ew i ing ules on mul ise s
o objec s, ha ep esen chemical subs ances, and s ings, ha ep esen he o ganisa ion o genes on he genome. The
inhe en andomness in biological phenomena is cap u ed by using s ochas ic s a egies. Mo eo e , he key ea u e o
P sys ems is he so called memb ane s uc u e which ep esen s he compa men alisa ion o he s uc u al o ganisa ion
o he cells, and whe e one can ake in o accoun he ole played by memb anes in he unc ioning o he sys em; o
ins ance, di usion, selec i e up ake o molecules om he en i onmen and signalling a he cell su ace.
In his pape we will p esen P sys ems as a eliable ool o he speci ica ion and simula ion o cellula p ocesses
wi hin he amewo k o Sys ems Biology. We will use a s a egy, based on he well known Gillespie’s algo i hm
bu unning on mo e han one compa men , called Mul i-compa men al Gillespie Algo i hm o he e olu ion o ou
sys ems.
The pape is o ganised as ollows. In he nex sec ion we p esen P sys ems as a speci ica ion language o cellula
p ocesses. In Sec ion 3 he Mul icompa men al Gillespie’s Algo i hm is desc ibed. A b ie desc ip ion o he gene
exp ession con ol in he Lac Ope on is gi en in Sec ion 4. The nex sec ion consis s o ou model o he Lac Ope on.
In Sec ion 6some esul s a e discussed. Finally, conclusions a e p esen ed in he las sec ion.
2. P Sys ems as a Specifica ion Language o Cellula P ocesses
In he s uc u e and he unc ioning o cells, memb anes play an essen ial ole. Cells a e sepa a ed om he
en i onmen by means o a skin memb ane, and hey a e in e nally compa men alised by memb anes.
The p ocesses aking place inside compa men s o on he memb anes delimi ing hem a e inhe en ly disc e e.
Chemical eac ions can be seen as ew i ing ules on mul ise s o objec s whe eby some objec s ( eac an s) a e eplaced
wi h o he s (p oduc s); and gene ic p ocesses like ansc ip ion and ansla ion can be seen as ew i ing ules on s ings
whe eby a subs ing (si e) is eplaced wi h ano he one.
Inspi ed by hese biological ea u es P˘
aun (2000) in oduced P sys ems as an uncon en ional model o compu a ion;
o de ails and upda ed in o ma ion on P sys ems we e e o P˘
aun (2002).
A P sys em is usually de ined as a hie a chical a angemen o a numbe o memb anes iden i ying a co esponding
numbe o egions inside he sys em. These egions a e associa ed o a ini e mul ise o objec s and a ini e se o ules.
We will also associa e a ini e mul ise o s ings o he memb anes ep esen ing he gene ic in o ma ion encoded in
DNA and RNA.
In wha ollows we gi e a p ecise de ini ion o he main componen s o a P sys em.
A P sys em is a cons uc
=(p o ,
dna,
na,L,μ,M
1,M
2,...,M
n,R
1,...,R
n)
whe e:
Fig. 1. Memb ane s uc u e.
•p o is a ini e alphabe o symbols ep esen ing p o eins;
•dna is a ini e alphabe o symbols ep esen ing DNA si es;
• na is a ini e alphabe o symbols ep esen ing RNA si es;
•Lis a ini e alphabe o symbols ep esen ing labels o he compa men s;
•μis a memb ane s uc u e con aining n≥1 memb anes iden i ied wi h numbe s om 1,...,nlabelled wi h elemen s
om L. Two memb anes wi h he same label will ep esen wo compa men s o he same ype.
Fo mally, a memb ane s uc u e is de ined as a hie a chical a angemen o memb anes whe e all he memb anes bu
one mus be included in a unique main memb ane, which de ines he bounda y o he sys em. The memb ane s uc u e
can be ep esen ed as a oo ed ee, whe e he nodes a e called memb anes, he oo is called skin, and he ela ionship
o a memb ane being inside ano he one is ep esen ed by he ela ionship o he node being he descenden o ano he
one.
We can ep esen a memb ane s uc u e using Venn diag ams.
In his pape , and in some ecen a emp s o model cell p ocesses (Che uku e al., 2007; P´
e ez-Jim´
enez and Rome o-
Campe o, 2006), a memb ane de ines an homogeneous and well s i ed egion; and so i will no necessa ily co espond
wi h a biological memb ane. Speci ically, in he sys ems we a e going o s udy he e a e biological memb anes ha
can be conside ed egions, whe e p o eins, ecep o s, channels, e c.... a e loca ed. In his case we will use a egion
delimi ed by wo memb anes, ins ead o a single memb ane, o speci y and simula e he p ocesses, aking place on he
memb ane su ace.
Fo ins ance, i in he sys em we a e s udying he e a e some ecep o s placed on he cell su ace in ol ed in
signalling ansduc ion we will use memb anes o ep esen his h ee di e en egions, namely, he en i onmen (e),
he cell su ace (s) and he cy oplasm (c). This memb ane s uc u e can be ep esen ed using a Venn diag am as in
Fig. 1.
•Mi=(li,w
i,s
i), o each 1 ≤i≤n, is he ini ial con igu a ion o memb ane iwi h li∈Li s ype, wi∈∗
p o a ini e
mul ise o objec -p o eins and siis a ini e mul ise o s ings o e dna;
•Ri, o each 1 ≤i≤n, is a ini e se o ules associa ed o memb ane i.
Rules o many di e en o ms ha e been conside ed o P sys ems in o de o encode he ope a ion o modi ying he
objec s inside he memb ane, o mo ing objec s om one egion o he o he , dissol ing, c ea ing, di iding memb anes,
e c. He e we will also conside ules wo king on s ings ep esen ing DNA o RNA.
The gene al o m o he ules we a e going o use is one o he ollowing:
•P o ein–p o ein in e ac ion ules:
u[ ]l→u[ ]l
whe e u, , u,
a e he mul ise s o objec s om p o and lis a label om L.
These ules a e mul ise ew i ing ules ha ope a e on bo h sides o he memb anes, ha is, a mul ise uplaced
ou side a memb ane labelled by land a mul ise placed inside he same memb ane can be simul aneously eplaced
by a mul ise uand a mul ise , espec i ely. In his way, we a e able o cap u e in a concise way he ea u es o
bo h communica ion ules and ans o ma ion ules. These ules a e e e ed o as bounda y ules in Be na dini and
Manca (2002).
•Gene ic ules:
[u,s]l→[u,s]l
whe e u, ua e he mul ise s o objec s om p o ,s, s he s ings o e dna ∪ na and lis a label om L.
These ules ope a e on bo h objec s and s ings, ha is, a mul ise uplaced inside a memb ane labelled wi h lwill
be eplaced wi h a mul ise u; simul aneously swill be ew i en by son a s ing placed inside he same memb ane
and ha ing sas subs ing.
We will also associa e o each ule a ini e se o a ibu es which a e mean o cap u e he quan i a i e aspec s
ha a e o en necessa y o cha ac e ise he eali y o he phenomenon o be modelled like kine ic o s ochas ic
cons an s.
In wha ollows we discuss in mo e de ail he speci ic ules we will use o model cellula p ocesses.
2.1. P o ein–P o ein In e ac ion Rules
– T ans o ma ion, complex o ma ion and dissocia ion ules:
[a]l→[b]l
[a, b]l→[c]l
whe e a, b, c ∈p o and l∈L
[a]l→[b, c]l
These ules a e used o speci y chemical eac ions aking place inside a compa men o ype l∈L, mo e speci ically
hey ep esen he ans o ma ion o ain o b, he o ma ion o a complex c om he in e ac ion o aand band he
dissocia ion o a complex ain o band c.
– Di using in and ou :
[a]l→a[]l
whe e a∈p o and l∈L
a[]l→[a]l
When chemical subs ances mo e o di use eely om one compa men o ano he we use hese ypes o ules,
whe e amo es om o o a compa men o ype l.
– Binding and debinding ules:
a[b]l→[c]l
whe e a, b, c ∈p o and l∈L
[a]l→b[c]l
Using ules o he i s ype we can speci y eac ions consis ing in he binding o a ligand swimming in one
compa men o a ecep o placed on he memb ane su ace o ano he compa men . The e e se eac ion, debinding
o subs ance om a ecep o , can be desc ibed as well using he second ule.
– Rec ui men and eleasing ules:
a[b]l→c[]l
whe e a, b, c ∈p o and l∈L
c[]l→a[b]l
Wi h hese ules we ep esen he in e ac ion be ween wo chemicals in di e en compa men s whe eby one o
hem is ec ui ed om i s compa men by a chemical on he o he compa men , and hen he new complex emains
in he la e compa men . In a eleasing ules a complex, c, loca ed in one compa men can dissocia e in o aand b,
emaining ain he same compa men as c, and bbeing eleased in o he o he compa men .
These ypes o ules ha e been used in Che uku e al. (2007); P´
e ez-Jim´
enez and Rome o-Campe o (2006) o model
signal ansduc ion pa hways consis ing o p o ein–p o ein in e ac ions. Nex , we desc ibe he ype o ew i ing ules
on s ings ha will be used o model gene ic eac ions. A p e ious a emp o model gene ic eac ions wi hin he
amewo k o P sys ems, bu using ew i ing ules only on mul ise s o objec s is p esen ed in Rome o-Campe o and
P´
e ez-Jim´
enez (in p ess). Ou a emp is no el in he sense ha we use ew i ing ules on bo h objec s and s ings as i
will be desc ibed in wha ollows.
2.2. Gene ic Rules
– Binding and debinding o a p o ein o speci ic si e on he DNA:
[p,si e]l→[si e]l
whe e p∈p o and si e, si e∈dna
[si e]l→[p, si e]l
A p o ein, p, can bind a egion o he DNA, si e, yielding a new egion si e. Re e sely pcan dissocia e om si e
p oducing he ee p o ein and a new egion si e.
– Polyme ase binding o a speci ic si e on he DNA:
[RNAP,si e]l→[si e RNAP]l
whe e RNAP ∈p o ep esen s he RNA polyme ase and si e ∈dna a p omo e in he DNA.
RNA polyme ase, RNAP, ecognises a speci ic si e on he DNA, binds he e and s a s ansc ip ion. When one
ule o his ype is applied si e is eplaced wi h si e RNAP in he co esponding s ing.
– T ansc ip ion:
[wRNAP si e]l→[si e wsi eRNAP]l
whe e w∈∗
na, si e ∈dna, si e∈ na and RNAP ∈p o ep esen s he RNA polyme ase.
RNAP ansc ibes DNA in o mRNA p oducing a s and o complemen a y RNA ha emains a ached o i
du ing he p ocess. The e o e, when RNAP ansc ibes a pa o he DNA, si e, i lea es i behind and a ached i s
complemen a y pa o RNA, si e, o he g owing mRNA w.
– Polyme ase dissocia ion when eaching a e mina ion si e in he DNA:
[wRNAP si e]l→[RNAP,si e,w]l
whe e w∈∗
na, si e ∈dna and RNAP ∈p o ep esen s he RNA polyme ase.
When he RNAP eaches a e mina ion si e i ends ansc ip ion by dissocia ing om he DNA and eleasing he
mRNA sequence ha can be, by ha ime, being ansla ed by some ibosomes.
– Binding o a ibosome o mRNA:
[Rib,si e]l→[Rib si e]l
whe e Rib ∈ na ep esen s a ibosome and si e ∈ na ep esen s a ibosome binding si e.

Sho ly a e RNAP s a s ansc ip ion and be o e i is o e , ibosomes bind o he g owing mRNA o s a
ansla ion. Using his ule he subs ing si e is eplaced wi h Rib si e consuming one objec Rib.
– T ansla ion:
[Rib si e]l→[si e Rib]l
whe e Rib ∈ na ep esen s a ibosome and si e ∈ na ep esen s he pa o he mRNA ha is going o be ansla ed.
Ribosomes ansla e mRNA in o a sequence o amino acids ha will yield a p o ein when he ibosome eaches
a e mina ion codon. Wi h hese ules we speci y how ibosomes mo e along mRNA, so he subs ing Rib si e is
eplaced wi h si e Rib ep esen ing ha si e has been ansla ed.1
– Dissocia ion o a ibosome om he RNA:
[Rib si e]l→[Rib,p,si e]l
whe e Rib ∈ na ep esen s a ibosome, si e ∈ na ep esen s a e mina ion si e on he mRNA and p∈p o
ep esen s he p o ein encoded on he pa mRNA ha has been ansla ed.
When a ibosome eaches a e mina ion codon i dissocia es om he mRNA eleasing he ansla ed p o ein. The
subs ing Rib si e igge s he end o ansla ion, hen he objec s Rib and pa e p oduced ep esen ing he dissocia ion
o a ibosome and he eleasing o he p o ein whe eas he subs ing Rib si e is eplaced wi h Si e on he co esponding
s ing ep esen ing he mRNA.
3. Mul icompa men al Gillespie’s Algo i hm
A undamen al esul o heo e ical s a is ical physics is he amous √nlaw, which s a es ha noise o luc ua ion
le el in a sys em a e in e sely p opo ional o he squa e oo o he numbe o pa icles. No e ha he c ucial ac o
is he numbe o pa icles, no he concen a ion. A sys em wi h ew pa icles in a e y small olume will esul in a
high concen a ion bu also a la ge ela i e noise. The e o e, cellula sys ems wi h low numbe o molecules show high
luc ua ions; and he de e minis ic and con inuous app oach o di e en ial equa ions is ques ionable. Ins ead s ochas ic
and disc e e app oaches, like Gillespie’s algo i hm, a e mo e accu a e.
Gillespie’s algo i hm (Gillespie, 1977)(see also (Gillespie, 2001, 2003) o some ecen imp o emen s) p o ides an
exac me hod o he s ochas ic simula ion o sys ems o bio-chemical eac ions; he alidi y o he me hod is igo ously
p o ed and i has been al eady success ully used o simula e a ious biochemical p ocesses (Meng e al., 2004). He e
we will use an ex ension o he classical Gillespie’s algo i hm called Mul i-compa men al Gillespie Algo i hm ha
was in oduced in P´
e ez-Jim´
enez and Rome o-Campe o (2006). This me hod is de eloped by aking in o accoun he
ac ha , wi h espec o he o iginal algo i hm whe e only one olume is s udied, in P sys ems we ha e a memb ane
s uc u e delimi ing di e en egions o compa men s. Each one o hese compa men s can be seen as a olume wi h
i s own se o ules, besides he applica ion o a ule inside a compa men can also a ec he con en o ano he one;
o example, he applica ion o a ule ha mo es objec s om one memb ane o ano he .
Speci ically, le =(p o ,
dna,
na,L,μ,M
1,M
2,...,M
n,R
1,...,R
n) be a P sys em as speci ied in he p e-
ious sec ion wi h memb anes Mi=(li,w
i,s
i) and ules Ri,1≤i≤n. Each ule, j, om he se Rio ules in
memb ane iwill be associa ed wi h an a ibu e cjwhich ep esen s a s ochas ic cons an , ha can be in e p e ed as
he a e age numbe o applica ions o he ule pe uni ime.
This cons an is used o compu e pj( he p obabili y o he ule j o be applied in he nex s ep o e olu ion) his
p obabili y is compu ed by mul iplying he s ochas ic cons an cj, speci ically associa ed as an a ibu e wi h ule j,by
he numbe o possible combina ions o he objec s and si es p esen on he le -side o he ules wi h espec o he
mul ise wiand he s ing si(o he mul ise wicon ained in he memb ane ou side memb ane i)− he cu en con en
o memb ane i(i).
Each memb ane iwill be conside ed as a compa men enclosing a olume, he e o e he index o he nex p og am
o be used inside memb ane iand i s wai ing ime will be compu ed using he classical Gillespie’s algo i hm which we
1No e ha in con as o ansc ip ion i is no necessa y o speci y he g owing sequence o amino acids since we a e no in e es ed in s udying
he p ocesses ha ake place on i .
ecall below:
(1) calcula e a0=pj, o all j∈Ri;
(2) gene a e wo andom numbe s 1and 2uni o mly dis ibu ed o e he uni in e al (0,1);
(3) calcula e he wai ing ime o he nex eac ion as τi=(1/a0)ln(1/ 1);
(4) ake he index j, o he p og am such ha j−1
k=1pk<
2a0≤j
k=1pk;
(5) e u n he iple (τi,j,i).
No ice ha he la ge he s ochas ic cons an o a ule and he numbe o occu ences o he objec s and si es placed
on he le -side o he ule inside a memb ane a e, he g ea e he chance ha a gi en ule will be applied in he nex
s ep o he simula ion. The e is no cons an ime-s ep in he simula ion. The ime-s ep is de e mined in e e y i e a ion
and i akes di e en alues depending on he con igu a ion o he sys em.
Nex , he Mul i-compa men al Gillespie’s Algo i hm is desc ibed in de ail:
•Ini ialisa ion
◦se ime o he simula ion =0;
◦ o each memb ane icompu e a iple (τi,j,i) by using he p ocedu e desc ibed abo e; cons uc a lis con aining
all such iples;
◦so his lis o iples (τi,j,i) in inc easing o de acco ding o τi;
•I e a ion
◦ex ac he i s iple (τm,j,m) om he lis ;
◦se ime o he simula ion = +τm;
◦upda e he wai ing ime o he es o he iples in he lis by sub ac ing τm;
◦apply he ule jin memb ane ionly once changing he numbe o objec s and si es in he memb anes a ec ed by
he applica ion o he ule;
◦ o each memb ane ma ec ed by he applica ion o he ule emo e he co esponding iple (τ
m,j,m
) om
he lis ;
◦ o each memb ane ma ec ed by he applica ion o he ule j e- un he Gillespie algo i hm o he new con ex
in m o ob ain (τ
m,j,m
), he nex ule j, o be used inside memb ane mand i s wai ing ime τ
m;
◦add he new iples (τ
m,j,m
) in he lis and so his lis acco ding o each wai ing ime and i e a e he p ocess.
•Te mina ion
◦Te mina e simula ion when ime o he simula ion eaches o exceeds a p ese maximal ime o simula ion.
The e o e, in his app oach, he wai ing ime compu ed by he Gillespie’s algo i hm is used o selec he memb anes
which a e allowed o e ol e in he nex s ep o compu a ion. Speci ically, in each s ep, he memb anes associa ed o
ules wi h he same minimal wai ing ime a e selec ed o e ol e by means o he co esponding ules. Mo eo e , since
he applica ion o a ule can a ec mo e han one memb ane a he same ime (e.g., some objec s may be mo ed om
one place o ano he ), we need o econside a new ule and wai ing ime o each one o hese memb anes by aking
in o accoun he new dis ibu ion o objec s and s ings inside hem. No e ha in his poin ou app oach di e s om
S undzia and Lumsden (1996) whe e only one ule is applied a each s ep wi hou aking in o accoun he es o ules
ha a e wai ing o be applied in o he memb anes, nei he i is conside ed he dis up ion ha he applica ion o one
ule can p oduce in a ious memb anes.
This algo i hm has been implemen ed using Scilab, a scien i ic so wa e package o nume ical compu a ions
p o iding a powe ul open compu ing en i onmen o enginee ing and scien i ic applica ions.
In he ollowing wo sec ions we b ie ly desc ibe he gene exp ession con ol in he Lac Ope on in E. coli and we
p esen a model using he o malism discussed in he p e ious sec ion.
4. Gene Exp ession Con ol in he Lac Ope on
Many o he genes in E. coli a e exp essed cons i u i ely; ha is, hey a e always u ned “on”. O he s, howe e , a e
ac i e only when hei p oduc s a e needed by he cell, so hei exp ession mus be egula ed. The mos di ec way
Fig. 2. Lac Ope on.
o con ol he exp ession o a gene is o egula e i s a e o ansc ip ion; ha is, he a e a which RNA polyme ase
ansc ibes he gene in o molecules o messenge RNA (mRNA).
Adding a new subs a e o he cul u e medium may induce he o ma ion o new enzymes capable o me abolising
ha subs a e. An example o his phenomenon happens when we ake a cul u e o E. coli ha is eeding on glucose and
ans e some o he cells o a medium con aining lac ose ins ead, a e ealing sequence o e en s akes place. A i s
he cells a e quiescen : hey do no me abolise lac ose, hei o he me abolic ac i i ies decline and cell di ision ceases.
Soon, howe e , he cul u e begins g owing apidly again wi h he lac ose being apidly consumed. Du ing he quiescen
in e al, he cells began o p oduce h ee enzymes ha hey had no been p oducing be o e: a pe mease, LacY, ha
anspo s lac ose ac oss he plasma memb ane om he cul u e medium in o he in e io o he cell; ␤-galac osidase
which hyd olyses lac ose in o glucose and galac ose, and a ansace ylase, LacA, whose unc ion is s ill unce ain.
The genes encoding hese p o eins and o he p o eins in ol ed in he egula ion o hei ansc ip ion a e loca ed on
a egion o he E. coli genome called Lac Ope on (Fig. 2).2
The gene lacI encodes a p o ein called LacI ha ac s as a ep esso . The lac ep esso is made up o ou iden ical
polypep ides ( he p o ein p oduc o he gene lacI). Pa o his molecule has a si e ha enables i o ecognise and bind
o 24 base pai s o he lac p omo e called he lac ope a o , op, p e en ing he RNA polyme ase om ansc ibing he
s uc u al genes lacZ, lacY and lacA ha encode ␤-galac osidase, he pe mease and he ansace ylase, espec i ely.
None heless, some imes he ep esso d ops om he p omo e allowing ansc ip ion a basal a e. Besides, he
ep esso con ains ano he si e whe e allolac ose, a p oduc o he eac ion o lac ose wi h ␤-gallac oside, can bind
p oducing a con o ma ional change. As a esul o his change, i can no longe bind o he ope a o egion and alls
o . RNA polyme ase can hen bind o he p omo e and ansc ibe he lac genes.
Thus, when lac ose is added o he cul u e medium, i causes he ep esso o be eleased om he ope a o so RNA
polyme ase can ansc ibe he h ee s uc u al genes o he ope on in o a single molecule o messenge RNA. Ha dly
does ansc ip ion begin be o e ibosomes a ach o he g owing mRNA molecule and mo e down i o ansla e i in o
he h ee p o eins. T ansc ip ion and ansla ion a e concu en p ocesses in bac e ia which play an impo an ole in
gene exp ession con ol.
Absence o ac i e lac ep esso is essen ial bu no su icien o e ec i e ansc ip ion o he Lac Ope on. The
p esence o glucose in he cul u e medium, e en in he p esence o lac ose, seems o ep ess o inhibi he syn hesis o
␤-galac osidase. The molecula mechanism o his e ec is called ca aboli e ep ession.
Ca aboli e ep ession is media ed h ough he e ec s ha glucose anspo in o he cell has on he in e nal concen-
a ion o cyclic AMP (cAMP). I glucose is abundan in he g ow h medium i will be anspo ed in o he cell by
he ac ion o he glucose anspo sys em. As i is being anspo ed, glucose is phospho yla ed wi h he phospha e
g oup being dona ed by a componen o he anspo sys em called EIIA ∼P. The same componen also ac i a es he
enzyme, adenyla e cyclase (AC). As long as he componen is pa icipa ing in glucose anspo , i is no able o ac i a e
adenyla e cyclase. The esul is ha as glucose is anspo ed in o he cell, he concen a ion o cAMP alls (because
adenyla e cyclase is no being ac i a ed o syn hesise any mo e). I he e is li le o no glucose in he g ow h medium, he
glucose anspo sys em is no ope a ional. The phospha e dono componen is now ee o ac i a e adenyla e cyclase.3
2An ope on is a g oup o genes physically linked on he ch omosome and unde he con ol o he same p omo e s. In an ope on, he linked genes
gi e ise o a single mRNA ha is ansla ed in o he di e en gene p oduc s. This ype o mRNA is called a polycis onic mRNA.
3EIIA, he non-phospho yla ed s a e o EIIA ∼P, inhibi s he pe mease in ol ed in he up ake o lac ose inside he bac e ial cell p e en ing
lac ose om en e ing he cy oplasm.
The esul is ha in he absence o glucose, he concen a ion o cAMP ises. Thus he e is an in e se ela ionship
be ween he ex e nal concen a ion o glucose and he cy oplasmic concen a ion o cAMP. As one ises, he o he alls.
The e o e, when glucose is sca ce in he medium, cAMP is abundan in he cy oplasm and i can be bound by he
cAMP ecep o p o ein (CRP), which is also known as ca aboli e ac i a o p o ein (CAP). As i s name sugges s, his
p o ein is esponsible o media ing he phenomenon o ca aboli e ep ession h ough i s abili y o ac i a e ansc ip ion.
The complex CRP–cAMP2binds o he Lac Ope on jus ups eam o he p omo e . In his posi ion i can assis RNA
polyme ase o bind by di ec p o ein–p o ein con ac s inc easing he a e o ansc ip ion hugely.
Summing up, he lac ose ope on is subjec o bo h nega i e and posi i e con ol. The lac ep esso , LacI, nega i ely
egula es exp ession and, he ac i a o , cAMP–CRP2, posi i ely ac i a es exp ession.
The e a e, as a esul , ou basic s a es o exp ession o he Lac Ope on:
•No glucose and no lac ose
Unde hese condi ions, he e will be a la ge numbe o cAMP molecules in he cell and CRP–cAMP2will be
bound a i s binding si e ups eam o he lac p omo e . I will assis RNA polyme ase o bind o he p omo e bu i
will no ac i a e ansc ip ion because he lac ose ep esso will emain bound o he ope a o si e since he e is no
induce , allolac ose, p esen .
The e will be essen ially no ansc ip ion o he Lac Ope on.
This makes physiological sense. Wi hou suga subs a es he cell canno ca y ou much me abolism; howe e ,
i emains poised o use wha e e i can whene e i can. In his case, i lac ose does become a ailable, he cell can
and will immedia ely espond because lac ose pe mease will anspo he lac ose in o he cell and RNA polyme ase
is posi ioned o s a he exp ession o ␤-galac osidase so ha he lac ose can be u ilized immedia ely.
•Glucose p esen bu no lac ose
Unde hese condi ions, he e will be a low numbe o cAMP molecules in he cell so CRP–cAMP2will no be
bound a he lac p omo e . In addi ion, he ac i i y o lac ose pe mease will be inhibi ed.
The e will be no ansc ip ion o he Lac Ope on.
This also makes physiological sense. As long as glucose is p esen in he g ow h medium he e is li le need o
me abolise lac ose and since lac ose is no p esen he e is no need o anspo lac ose in o he cell o o exp ess he
genes o he Lac Ope on.
•Glucose and lac ose p esen
Unde hese condi ions, he e will be a low numbe cAMP molecules in he cell so CRP–cAMP2will no be bound
a he lac p omo e . Lac ose pe mease will be inhibi ed bu some lac ose will s ill en e he cell.
The e will be a low le el ansc ip ion o he Lac Ope on.
Again, his makes physiological sense. As long as glucose is p esen in he g ow h medium he e is li le need
o me abolise lac ose. Howe e , since lac ose is now p esen , he cell would be oolish o igno e a suga supply
comple ely. The Lac Ope on will be induced bu , since CRP is no bound, he amoun o ansc ip ion is ela i ely
low.
•No glucose bu abundan lac ose
Unde hese condi ions, he e will be a la ge numbe o cAMP molecules in he cell so CRP–cAMP2will be
bound a he lac p omo e . Lac ose pe mease is no inhibi ed, so i will anspo he lac ose in o he cell.
The e will be maximal ansc ip ion o he Lac Ope on.
This also makes physiological sense. Wi h lac ose as he sole suga sou ce, he cell mus use e e y a ailable
molecule o i s own bene i . Thus he lac ose pe mease anspo sys em will anspo lac ose in o he cell and he
Lac Ope on will be bo h induced and ac i a ed.
The p esence o wo sepa a e con ol sys ems allows he cell o espond mo e sensi i ely o he needs imposed by
changing g ow h condi ions. Many bac e ial ope ons ha e dual con ol sys ems. The de ails a e di e en in he di e en
cases, howe e (P shane, 2004; P ashne and Gann, 2002).
5. A Model o he Lac Ope on
In his sec ion we p esen a model o he gene exp ession con ol in he Lac Ope on using P sys ems. We will s udy
he beha iou o sys em o di e en ini ial condi ions wi h/wi hou glucose and wi h/wi hou lac ose.
Obse e ha because CRP–cAMP2is bound o he CAP si e, RNAP (in bold) will be eady o s a ansc ip ion
whene e he ep esso d ops om he ope a o . No e ha on he p e ious s ing he e a e wo polyme ases, highligh ed
in bold, ansc ibing he ope on. This will p oduce a sligh inc ease in he exp ession o he genes encoded in he Lac
Ope on. This makes physiological sense, wi h no suga in he en i onmen he bac e ium se s he ope on such ha i
can espond immedia ely and e icien ly o he p esence o lac ose.
6.2. No Glucose bu Abundan Lac ose
As men ioned be o e he absence o glucose p oduces a high numbe o ac i a o s and so a CRP–cAMP2molecule
will be bound o he CAP si e ac i a ing he ec ui men o he RNAP. Besides when lac ose is abundan in he
en i onmen i will be anspo ed by he pe mease LacY, ha is exp ess a a basal le el, in o he cy oplasm. ␤-
Galac osidase is also p esen a a basal le el in he cy oplasm and as soon as lac ose is p esen i s a s o clea e
i in o galac ose and glucose, and also occasionally allolac ose appea s as a p oduc o he in e ac ion be ween ␤-
galac osidase and lac ose. Allolac ose ac s as an induce binding o he ep esso LacI and p e en ing i om binding
o he ope a o . In Fig. 4 i is depic ed how he numbe o ac i e ep esso s is apidly inhibi ed when lac ose is
abundan .
Unde hese condi ions he Lac Ope on will be bo h induced (no ep esso will be bound o he ope a o ) and
ac i a ed ( he ac i a o will be bound o he CAP si e). The e o e, he con igu a ion o he swi ch will be capCRP–cAMP2
op and he genes encoded in he ope on will be ansc ibed massi ely, as i can be deduced by he numbe o RNAP
ansc ibing he ope on (Fig. 4). This will esul in a d as ic inc ease in he numbe o ␤-galac osidase and LacY
molecules, see Fig. 5.
Fig. 4. Numbe o ac i e ep esso s and RNAP.

Fig. 5. Numbe o ␤-galac osidase and pe mease LacY o e ime.
6.3. Glucose P esen bu No Lac ose
Unde hese condi ions, EIIA ∼P is deple ed apidly om he cy oplasm by he glucose anspo sys em and he
ac i i y o AC is ep essed p oducing a low numbe o ac i a o s, see Fig. 6. The e o e, no CRP will no be bound a
he lac p omo e . In addi ion, he ac i i y o lac ose pe mease will be inhibi ed.
Since he e is no lac ose in he en i onmen he ep esso will be ac i e and bound o he ope a o p oducing
only ansc ip ion o he Lac Ope on a a basal a e, as i can be seen in he low numbe o ac i e polyme ases in
Fig. 7.
This makes physiological sense. As long as glucose is p esen in he g ow h medium he e is li le need o me abolise
lac ose and he lac ose ope on is swi ch o .
6.4. Glucose and Lac ose P esen
Again he p esence o glucose p oduces a low numbe o ac i a o s and he Lac Ope on will no be ac i a ed by he
binding o a CRP–cAMP2molecule o he CAP si e. The e o e, e en in he p esence o lac ose he genes encoded in
he Lac Ope on will be o ,5 he e will be li le up ake o lac ose om he en i onmen .
Obse e in Fig. 8 how he numbe o glucose molecules dec eases in he en i onmen whe eas lac ose emains
almos cons an .
5This phenomenon is known as ca aboli e ep ession.
Fig. 6. Numbe o ac i e EIIA ∼P and ac i a o CRP–cAMP.
None heless, some lac ose will be p esen in he cy oplasm which will p oduce allolac ose able o inhibi he
ep esso o some ex end p oducing a low ansc ip ion o he ope on. This can be seen in Fig. 9 whe e he numbe o
galac osidase and LacY s a o inc ease slowly.
Again, his makes physiological sense. As long as he bac e ium can me abolise glucose he e is li le need
o me abolise lac ose. Howe e , since lac ose is now p esen , he cell would no igno e a suga supply com-
ple ely. The Lac Ope on will be induced bu , since CRP is no bound, he amoun o ansc ip ion is ela i ely
low.
Fig. 7. Numbe o polyme ases ansc ibing he Lac Ope on in p esence o glucose.
Fig. 8. Glucose and lac ose molecules in he en i onmen .
Fig. 9. Numbe o ␤-galac osidase and pe mease LacY o e ime.
7. Conclusions
In his pape we ha e used P sys ems as a o mal amewo k o he speci ica ion and simula ion o biological
sys ems in ol ing p o ein–p o ein and p o ein–DNA in e ac ions. Ou app oach akes in o accoun he key ole played
by memb anes in he s uc u e and unc ioning o he cells and he disc e e and concu en cha ac e o p ocesses in
bio-sys ems.
The models speci ied using P sys ems can e ol e using di e en s a egies/algo i hms. In his wo k we ha e used he
Mul i-compa men al Gillespie’s Algo i hm. This s a egy equi es much compu a ional esou ces and so he au ho s
will s udy he possibili y o adap imp o ed e sion o Gillespie’s algo i hm, like Gillespie (2001, 2003), in o de o
de elop mo e e icien s a egies o he e olu ion o P sys ems.
A model o he gene exp ession con ol in he Lac Ope on in E. coli has been also p esen ed as a case s udy. We ha e
s udied he beha iou o he sys em o di e en en i onmen al condi ions o see how he sys em is able o sense he
p esence o di e en subs a es (glucose and lac ose) using he cell su ace and eac acco ding o hem by syn hesising
in he cy oplasm he enzymes necessa y o consume hem. No e ha we ha e modelled explici ly ansc ip ion and
ansla ion as ew i ing and concu en p ocesses on s ings, see s ing on page 21. The delay be ween he sensing o
he signal and he exp ession o di e en genes is no modelled explici bu eme ge as a consequence o he o mula ion
o ou app oach.
Ou esul s ag ee well wi h expe imen al obse a ions and esul s ob ained using o he app oaches. This shows he
eliabili y o P sys ems as compu a ional modelling ools o p oduce pos dic ion and pe haps as he ield e ol es hey
will be able o p oduce plausible p edic ions.
Acknowledgemen s
This wo k is suppo ed by Minis e io de Ciencia y Tecnolog´
ıa o Spain, by Plan Nacional de I+D+I(2005–2007)
(TIN2005-09345-C04-01), co inanced by FEDER unds, and by a FPU ellowship om he Minis e io de Ciencia y
Tecnolog´
ıa o Spain.
Re e ences
Be na dini, F., Manca, V., 2002. P sys ems wi h bounda y ules. In: P˘
aun, Gh., Rozenbe g, G., Salomaa, A., Zand on, C. (Eds.), Memb ane
Compu ing, In e na ional Wo kshop, ol. 2597 o Lec u e No es in Compu e Science. Sp inge , Cu ea de A ges, Romania, pp. 107–118.
Bianco, L., Fon ana, F., Manca, V., 2005. P sys ems and he modelling o biochemical oscilla ion. In: F eund, R., P˘
aun, Gh., Rozenbe g, G.,
Salomaa, A. (Eds.), Memb ane Compu ing, In e na ional Wo kshop, ol. 3850 o Lec u e No es in Compu e Science. Sp inge , Vienna, Aus ia,
pp. 199–208.
Che uku, S., P˘
aun, A., Rome o Campe o, F.J., P´
e ez Jim´
enez, M.J., Iba a, O.H., 2007. Simula ing FAS induced apop osis by using P sys ems. P og.
Na . Sci. 17 (4), 424–431.
Ciobanu, G., P˘
aun, Gh., P´
e ez-Jim´
enez, M.J. (Eds.), 2006. Applica ions o Memb ane Compu ing, Na u al Compu ing Se ies. Sp inge , Be lin.
Gillespie, D.T., 1977. Exac s ochas ic simula ion o coupled chemical eac ions. J. Phys. Chem. 81 (25), 2340–2361.
Gillespie, D.T., 2001. App oxima e accele a ed s ochas ic simula ion o chemically eac ing sys ems. J. Chem. Phys. 115 (4), 1716–1733.
Gillespie, D.T., 2003. Imp o ed leap-size selec ion o accele a ed s ochas ic simula ion. J. Chem. Phys. 119 (16), 8229–8234.
Hube , R.E., Wallen els, K., Ku z, G., 1975. Ac ion o ␤-galac osidase on allolac ose. Can. J. Biochem. 53, 1035–1038.
Kennell, D., Riezman, H., 1977. T ansc ip ion and ansla ion ini ia ion equencies o he Esche ichia coli Lac Ope on. J. Mol. Biol. 114, 1–21.
Kie zek, A.M., Zaim, J., Zielenkiewicz, P., 2001. The e ec o ansc ip ion and ansla ion ini ia ion equencies on he s ochas ic luc ua ionsin
p okayo ic gene exp ession. J. Biol. Chem. 276 (11), 8165–8172.
Lolkema, J.S., Ca asco, N., Kaback, R., 1991. Kine ic analysis o lac ose exchange in p o eoliposomes econs i u ed wi h pu i ied lac pe mease.
Biochemis y 30, 1284–1290.
Meng, T.C., Somani, S., Dha , P., 2004. Modelling and simula ion o biological sys ems wi h s ochas ici y. In Silico Biol. 4, 0024.
P˘
aun, Gh., 2000. Compu ing wi h memb anes. J. Compu . Sys . Sci. 61 (1), 108–143.
P˘
aun, Gh., 2002. Memb ane Compu ing. An In oduc ion. Sp inge , Be lin.
Pescini, D., Besozzi, D., Mau i, G., Zand on, C., 2006. Dynamical p obabilis ic P sys ems. In . J. Found. Compu . Sci. 17 (1), 183–204.
P´
e ez-Jim´
enez, M.J., Rome o-Campe o, F.J., 2006. P sys ems, a new compu a ionl modelling ool o sys ems biology. T ans. Compu . Sys . Biol.
VI 4220, 176–197.
P ashne, M., 2004. A Gene ic Swi ch. Phage Lambda Re isi ed, hi d ed. Cold Sp ing Ha bo Labo a o y P ess, New Yo k.
P ashne, M., Gann, A., 2002. Genes and Signals. Cold Sp ing Ha bo Labo a o y P ess, New Yo k.
Rome o-Campe o, F.J., P´
e ez-Jim´
enez, M.J. A model o he quo um sensing sys em in Vib io fische i using P sys ems. A i . Li e J., in p ess.
Rohwe , J.M., Meadow, N.D., Roseman, S., Wes e ho , H., 2000. Unde s anding glucose anspo by he bac e ial phosphoenolpy u a e: glycose
phospho ans e ase sys em on he basis o kine ic measu emen s in i o. J. Biol. Chem. 275 (45), 34909–34921.
S undzia, A.B., Lumsden, C.J., 1996. S ochas ic simula ion o coupled eac ion—di usion p ocesses. J. Compu . Phys. 127, 196–207.
Wong, P., Gladney, S., Keasling, J.D., 1997. Ma hema ical model o he Lac Ope on: induce exclusion, ca aboli e ep ession, and diauxic g ow h
on glucose and lac ose. Bio echnol. P og. 13, 132–143.