Towa ds a P Sys ems Pseudomonas Quo um
Sensing Model
Luca Bianco1, Da io Pescini2, Pe e Siepmann3, Na alio K asnogo 3,
F ancisco J. Rome o-Campe o4, and Ma ian Gheo ghe5
1Depa men o Compu e Science, Uni e si y o Ve ona
S ada Le G azie 15, 37134 Ve ona, I aly
[email p o ec ed]
2Dipa imen o di In o ma ica, Sis emis ica e Comunicazione
Uni e si `a degli S udi di Milano-Bicocca
Via Bicocca degli A cimboldi 8, 20126 Milano, I aly
[email p o ec ed]
3School o Compu e Science and In o ma ion Technology
Uni e si y o No ingham
Jubilee Campus, No ingham, NG81BB, UK
{Pe e .Siepmann, Na alio.K asnogo }@no ingham.ac.uk
4Resea 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 ille, A da. Reina Me cedes, 41012 Se illa, Spain
[email p o ec ed]
5Depa men o Compu e Science, The Uni e si y o Sheffield
Regen Cou , Po obello S ee , Sheffield S1 4DP, UK
[email p o ec ed]
Abs ac . Pseudomonas ae uginosa is an oppo unis ic bac e ium ha
exploi s quo um sensing communica ion o synch onize indi iduals in a
colony and his leads o an inc ease in he effec i eness o i s i ulence.
In his pape we de i ed a mechanis ic P sys ems model o desc ibe he
beha io o a single bac e ium and we discuss a possible app oach, based
on an e olu iona y algo i hm, o une i s pa ame e s ha will allow a
quan i a i e simula ion o he sys em.
1 In oduc ion
The quo um sensing is a pa icula o m o cell- o-cell communica ion in bac e ia
which exploi s he concen a ion o a pa icula molecule, called signal, o “sense”
he popula ion densi y o he colony. The quo um sensing egula o y ne wo k is
used by he indi iduals o he colony o collec i e synch oniza ion and he e o e
o a cohe en con ol o e he gene exp ession. In Pseudomonas ae uginosa his
mechanism is esponsible o he effec i eness o he i ulence o his bac e ium
[14,23,10,9]. In ac , a single bac e ium s a s o exp ess his i ulence ac o s
only when i senses ha he bac e ia popula ion has eached a ce ain h eshold
le el such ha he hos esponse will be inadequa e.
The ac i a ion o a complex cellula esponse is wha dis inguishes he quo um
sensing as a communica ion egula o y ci cui om o he densi y dependen
esponses such as he me aboliza ion o de oxifica ion o small molecules.
The simples quo um sensing ne wo k known in G am-Nega i e bac e ia is
also he fi s one e e disco e ed [25,17]. I has been ound in he Vib io fische i
bac e ium, also known as Pho obac e ium fische i and is nowadays conside ed
as he pa adigm o his cell communica ion p ocess. In his ne wo k wo p o eins
and one signalling molecule a e in ol ed. The Rp o einis a ansc ip ional egu-
la o , while he Ip o einis he syn hase o he signalling molecule, also e e ed
o as he au oinduce . An impo an ole is also played by he confinemen o
he bac e ial colony. The ac ha he au oinduce molecule is no dispe sed
in he en i onmen allows i s diffusion inside he indi iduals and he e o e i s
concen a ion sensing.
A low cell densi ies he Ip o einsyn hesizes he au oinduce a a basal a e
and he signal eely diffuses ou side he bac e ium. The concen a ion o he sig-
nal inside each bac e ium is inc eased by he combined effec o he confinemen
and he inc ease o he popula ion. A his poin , he binding o he Rp o ein
wi h he au oinduce becomes mo e likely. The binding o he signal molecules
ac i a es he Rp o ein ansc ip ional egula o . Since he I gene is he a ge
o he Rp o ein, he bac e ium s a s o p oduce mo e and mo e signal. The
egula ion ne wo k signal au oinduces i s ansc ip ion. In his way he high con-
cen a ion o he au oinduce coo dina es he ansc ip ion o all he genes ha
a e a ge o he Rp o ein.
The quo um sensing in Pseudomonas ae uginosa is mo e complex, ne e heless
in iguing, since his bac e ium uses wo diffe en quo um sensing sys ems which
in e ac wi h each o he .
The aim o his wo k is o p o ide a P sys em model [15,16] o he bac e ium
Pseudomonas ae uginosa quo um sensing ocusing on he communica ion mech-
anisms. The pa ame e s o he model will be uned using an e olu iona y al-
go i hm. Ou long e m aim is o ep oduce he cha ac e is ic beha io o he
quo um sensing in Pseudomonas ae uginosa, namely, he swi ch be ween wo
dis inc s able s eady solu ions: he fi s desc ibing he beha io o he non-
quo a ed bac e ium (i.e., wi h low le els o au oinduce ), he second modeling
i s quo a ed beha io (i.e., he beha io ob ained wi h high concen a ion o he
au oinduce molecule). Once he model will be en i ely defined se e al simula-
ions wi h diffe en s a egies [6,20,18] will be un.
Fi s o all, we add ess he modeling o he in e nal dynamics o one single
bac e ium, uning i s kine ic cons an s in a way ensu ing i s non-quo a ed be-
ha io . A a la e s age, we in end o exploi compa men aliza ion o P sys ems
o model a colony o bac e ia each o hem in e nally specified acco ding o he
same se o kine ic cons an s. In his espec we will ex end he cu en model
o a Popula ion P sys ems app oach [3] ha has been al eady used o exp ess
some aspec s o quo um sensing in bac e ium Pseudomonas ae uginosa [22] and
o sel -assembly p oblems [4].
2 An Ini ial Model
The fi s s age o ou in es iga ion is in ended o desc ibe he quo um sensing
ela ed ne wo k o each bac e ium o cap u e i s main ea u es in o a mecha-
nis ic model. The quo um sensing in e nal pa hway o each bac e ium is aken
om models discussed in [11,9] and a g aphical ep esen a ion o all elemen s
in ol ed in i , as well as some ele an ela ionships be ween hem, a e depic ed
in Figu e 1.
LasR
LasR
lasR lasI
LasI
3O
V
saL
RsaL
Fig. 1. The Pseudomonas quo um sensing model analyzed he e ( om [9]). No e ha
double a ows deno e e e sible eac ions, bold ones he deg ada ion p ocess and he
emp y ones he inhibi o y p ocess.
Acco ding o his model, he quo um sensing pa hway comp ises wo in e con-
nec ed signalling cascades. The main elemen s in ol ed in he fi s one a e p o-
eins LasR,RsaL,LasI (as well as he genes in ol ed in hei p oduc ion), he
au oinduce molecule 3-oxo-C12-HSL and he ac i e complex LasR-3-oxo-C12-
HSL. The key elemen s o he second sys em a e he p o eins RhlR and RhlI
(as well as he genes in ol ed in hei p oduc ion), he au oinduce molecule C4-
HSL and he ac i e complex RhlR-C4-HSL. The fi s one o he wo signalling
cascades is called las sys em because i was shown o egula e he exp ession
o LasB elas ase. This pa hway egula es o he i ulence ac o s such as LasA
p o ease, exo oxin A, alkaline p o ease A as well as he exp ession o a leas wo
genes o he xcp sec e o y pa hway. The las pa hway is posi i ely con olled by
GacA and V whe eas i is inhibi ed by RsaL ha , in u n, is posi i ely egu-
la ed by he ac i e complex LasR-3-oxo-C12-HSL and whose ole is o ep ess
he ansc ip ion o he lasI gene.
The second signalling sys em in ol ed in he model is named hl sys em be-
cause i con ols he exp ession o hamnolipid ia he p oduc ion o hlAB
ope on. The au oinduce molecule in his case is C4-HSL and he ac i e com-
plex is RhlR-C4-HSL. I has been shown ha his cascade is necessa y o he
p oduc ion o some i ulence ac o s like LasB elas ase and LasA p o ease, as
well as pyocyanin, cyanide and alkaline p o ease. Fo his eason his signalling
sys em is also known as sm ( i ulence seconda y me aboli es).
Al hough he co esponding au oinducing molecules a e highly selec i e (and
hus no in e changeable a all), se e al in e connec ions be ween he las and
he hl pa hways o he quo um sensing in Pseudomonas ae uginosa a e known.
One link be ween hem has been al eady men ioned and i is cons i u ed by
he LasB elas ase, ha needs bo h LasR-3-oxo-C12-HSL and RhlR-C4-HSL
o i s p oduc ion. Mo e in e es ingly, he las sys em is a a highe le el in he
hie a chical egula o y cascade, in ac LasR-3-oxo-C12-HSL can ac i a e he
exp ession o he hlR gene. In addi ion, he ac i e complex LasR-3-oxo-C12-
HSL canbind oRhlR p e en ing i o o m he complex RhlR-C4-HSL.
2.1 The Diffe en ial Equa ion Model
Many models o he quo um sensing in he Pseudomonas ae uginosa a e p e-
sen ed in li e a u e and usually hey app oach he phenomenon om wo diffe -
en angles. The fi s one desc ibes he colony beha io by summa izing indi idual
dynamics as a s a e change a oiding a p ecisely de ailed ep esen a ion o each
o he bac e ium quo um sensing ne wo ks [24,1]. The second one desc ibes in a
mo e de ailed ashion he quo um sensing pa hway o each bac e ium wi h he
pu pose o model he eme gen beha io o he whole colony [11].
We hink ha he P sys em amewo k is pa icula ly sui able o his second
app oach. In ac , he modula i y, he compa men aliza ion, he hie a chical
s uc u e and he ew i ing ules (all ea u es o P sys ems [16]) allow a con e-
nien desc ip ion o his eali y.
In [11] a model o he las signalling sys em has been de ised, bu no desc ip-
ion is gi en o he hl sys em. The g aphical desc ip ion o he quo um sensing
pa hway depic ed in Figu e 1 has been ansla ed in o he se o eigh diffe en ial
equa ions p esen ed in he nex page. The co espondence be ween diffe en ial
equa ion symbols and elemen s in he pa hway a e summa ized in Table 1.
The p oduc ion o he ac i a ed complex Pby means o he au oinduce and
he LasR p o ein (whose exp ession is gi en by he p oduc o he cons i u i e
elemen s concen a ions wi h a a e kRA:kRARA)isanexampleo howcoope -
a i e con ibu ions a e ob ained in he diffe en ial equa ions app oach by means
o he mass ac ion law.
Basal a es p oduc ions and deg ada ions a e also aken in o accoun , an ex-
ample o he o me being he k1 elemen gi ing he basal p oduc ion o LasR
p o ein (R), while an example o he la e is he deg ada ion o he ac i e com-
plex (P) ep esen ed by he elemen kPP. The p oduc ion o messenge RNAs
om he co esponding genes is modeled wi h a Michaelis-Men en-like dynam-
ics depending on he concen a ion o he p omo ing ac o , as i happens in
he case o he p oduc ion o lasR and saL mRNAs ( espec i ely and s), he
fi s modeled by V
P
K +Pand he second by Vs
P
Ks+P. The p oduc ion o lasI
mRNA (l) is also down- egula ed by he p esence o RsaL p o ein (S)and his
is modeled by Vl
P
Kl+P
1
KS+S, in which he Michaelis-Men en-like dynamics
is a enua ed by an in e sely p opo ional unc ion o he RsaL concen a ion.
dP
d =kRARA −kPP
dR
d =−kRARA +kPP−kRR+k1
dA
d =−kRARA +kPP+k2L−kAA
dL
d =k3l−kLL
dS
d =k4s−kSS
ds
d =Vs
P
Ks+P−kss
d
d =V
P
K +P−k + 0
dl
d =Vl
P
Kl+P
1
KS+S−kll+l0
(1)
Un o una ely, no alue is known o he 21 kine ic cons an s p esen in he se
o diffe en ial equa ions (1). To o e come his p oblem, in [11] se e al simpli ying
assump ions a e conside ed, ha lead o ewe equa ions and ewe pa ame e s
as well.
In he ollowing we will desc ibe a possible pa ame e es ima ion s a egy
o ackle his p oblem (see Sec ion 4). The idea is o elay o his diffe en ial
equa ions sys em as a “syn he ic bio-expe imen ” used o con on ou model o.
2.2 A Fi s P Sys ems Model
Se e al a emp s o simula e he quo um sensing in bac e ia a e p esen in P
sys ems li e a u e [5,19], bu , as a as we know, none o hem deals wi h he
Pseudomonas ae uginosa bac e ium.
He e we desc ibe a di ec P sys ems ansla ion o he diffe en ial equa ion
model p e iously discussed [11]. Fo mally, he Pseudomonas P sys em is
Π=(A, μ, w, R)

Table 1. Va iable-concen a ion co espondence be ween he diffe en ial o mula ion
and he g aphical desc ip ion o he quo um sensing model o Pseudomonas ae uginosa
( om [11])
Va iable Concen a ion
RLasR
A3-oxo-C12-HSL
PLasR-3-oxo-C12-HSL
LLasI
SRsaL
lasR mRNA
llasI mRNA
s saL mRNA
whe e:
–A={geneR,geneL,R,A,P,L,S, ,l,s}is he alphabe ;
–μ=[]
0is he memb ane s uc u e: since we add ess he single bac e ium
case, i con ains he cellula memb ane only;
–w=geneR geneL is he ini ial configu a ion ha comp ises only LasR and
LasI genes, hus is ep esen ed as he s ing;
–R={ 1,···,
18}is he se o he ules:
1:geneR −→ geneR +
2: −→ λ
3: −→ +R
4:P−→ P+
5:R+A−→ P
6:P−→ R+A
7:P−→ P+s
8:s−→ λ
9:S−→ λ
10 :s−→ s+S
11 :P−→ P+l
12 :l−→ l+L
13 :l−→ λ
14 :geneL −→ geneL +l
15 :L−→ λ
16 :L−→ L+A
17 :A−→ λ
18 :R−→ λ
No e ha , symbols in Aco espond o he a iables o he diffe en ial equa ion
and hei co espondence o he biological eali y is gi en in Table 1. Two new
elemen s (i.e., geneR and geneL) a e in oduced, which accoun o he genes
in ol ed in he basal p oduc ion o he LasR and LasI mRNAs.
Each one o he ules in Ris di ec ly ob ained om he diffe en ial desc ip ion
o he conside ed quo um sensing model. Fo examples, we can see ha ule 1
models he basal p oduc ion o he LasR mRNA, while ule 2exp esses i s
deg ada ion, mo eo e ules 5and 6desc ibe he e e sible eac ion o he
complex P o ma ion by s a ing om i s undamen al cons i uen s Rand A.
Due o he diffe en le el o abs ac ion in he ep esen a ion o diffe en
pa s o he model (as in he case o he Michaelis-Men en-like kine ics ha
a e modelled wi h a highe le el o abs ac ion han o he componen s o he
sys em), we canno di ec ly apply mechanis ic algo i hms [2] o his model. Fo
his eason, we will apply o his se o ules only he s a egy known as Me abolic
Algo i hm ( o de ails e e o [6]), whose simula ion esul s, oge he wi h some
nume ical solu ions o he se o diffe en ial equa ions (1), a e shown in Sec ion
2.3 o diffe en choices o pa ame e s.
The me abolic algo i hm simula ion needs o speci y a se o eac ion maps,
each one associa ed in a one- o-one manne o he ules o R. Reac ion maps [6]
a e unc ions defined o e he s a e o he sys em (i.e., mul iplici y o concen-
a ion o all elemen s o he sys em depending on he case), ha a e used by
he Me abolic algo i hm o alloca e objec s o ules. Fo example, as we will see
in a while, F 1, ha is he eac ion map o ule 1, is simply he cons an a e
o p oduc ion o LasR mRNA. We can ha e mo e complica ed eac ion maps,
as in he case o ule 4 ha akes in o accoun he Michaelis-Men en-like p o-
duc ion o he LasR mRNA elici ed by he LasR-3oxo-C12-HSL complex. As
in he case o he ules, ha speci y he physical in e ac ions and connec ions
be ween he elemen s o he modeled eali y, we can ob ain his in o ma ion om
he diffe en ial equa ion o mula ion. The se o eac ion maps employed in ou
simula ions a e he ollowing:
F 1= 0F 2=k
F 3=k1F 4=V
K +P
F 5=kRA F 6=kP
F 7=Vs
Ks+PF 8=ks
F 9=k4F 10 =kS
F 11 =Vl
(Kl+P)·(KS+S)F 12 =k3
F 13 =klF 14 =l0
F 15 =kLF 16 =k2
F 17 =kAF 18 =kR
(2)
No e ha all eac ion maps a e cons an apa om h ee o hem. We ha e al-
eady discussed he meaning o he eac ion map associa ed o ule 4; analogous
conside a ions hold o F 7as well. Mo e in e es ing is he eac ion map associ-
a ed o ule 11 ha akes in o accoun he inhibi o y effec o RsaL p o ein on
he p oduc ion o he lasI mRNA.
Rema kably, he me hod allows he cu en desc ip ion o diffe en pa s o
he sys em a diffe en abs ac ion le els; mo eo e i is s ill applicable i all
eac ion maps a e cons an , a condi ion equi ed by mechanis ic algo i hms.
In he ollowing some simula ion esul s a e shown, as well as he nume ical
solu ion o he diffe en ial equa ion sys em, o some chosen pa ame e s.
2.3 Simula ion Resul s
He e we show how he same model- eali y can be desc ibed wi h wo diffe en
app oaches. As men ioned be o e, we do no ha e p ecise alues o he model
pa ame e s. Fo his eason, as a fi s compa ison a emp , we make a comple ely
fic i ious choice o hem. As a u he wo k, we plan o adop some au oma ic
way o he pa ame e es ima ion (see Sec ion 4 o mo e de ails). The ini ial
choice o pa ame e s is he e shown, and all he subsequen changes o his ini ial
pa ame e se will be explici ly men ioned:
kRA =10 kP=2
kR=5 k1=1
k2=1 kA=1
k3=1 kL=1
k4=1 kS=1
Vs=1 Ks=1
ks=0.5V =1
K =1 k =1
0=1 Vl=1
Kl=1 kl=1
l0=1 KS=1
(3)
The lack o biological in o ma ion makes his choice comple ely a bi a y and
p e en s us o compu e he dynamics o he sys em by means o s ochas ic
algo i hms such as he Gillespie one [12,13], Dynamical P obabilis ic P Sys ems
[20] o he Mul i-compa men al Gillespie [18].
In his sec ion we compa e he dynamics gene a ed by he me abolic algo-
i hm wi h he solu ions ob ained o he co esponding diffe en ial equa ion
sys em. Figu e 2 depic s he case in which pa ame e s a e chosen acco ding o
(3). The dynamics o each species eaches a s eady s a e in bo h app oaches,
bu he ela i e posi ion o he species is diffe en and his leads o wo dis inc
sys em dynamics. Mo eo e , he ime o he wo sys ems diffe s; in he solu ion
o he diffe en ial equa ion sys em his is measu ed in a bi a y uni s (due o
he a bi a y choice o pa ame e s), while in he model based on P sys ems he
ime is measu ed in s eps o sys em e olu ion. In Figu e 3 he choice o Vl=0
swi ches off ule 11 o he P sys em model and in his case he esul s o he
wo diffe en app oaches quali a i ely ma ch each o he . Finally, he las choice
o pa ame e s is aimed a ob aining a quo um sensing consis en beha io , ha
is, in he case o a single bac e ium in he en i onmen i should no quo a e
and hus he concen a ion o he complex Pshould each he basal a e. Ac-
co dingly, we se KRA o he alue 0.1. In his case, depic ed in Figu e 4, he
dynamics p oduced by he wo app oaches is quali a i ely simila again.
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30
P
R
A
L
S
s
l
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5000 10000 15000 20000 25000 30000
P
R
A
L
S
s
l
Fig. 2. Resul s o he quo um sensing model wi h pa ame e s showed in (3) using
ODE app oach (le ) and me abolic algo i hm ( igh )
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30
P
R
A
L
S
s
l
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5000 10000 15000 20000 25000 30000
P
R
A
L
S
s
l
Fig. 3. Resul s o he quo um sensing model wi h Vl= 0 using ODE app oach (le )
and me abolic algo i hm ( igh )
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
P
R
A
L
S
s
l
0
0.2
0.4
0.6
0.8
1
1.2
0 5000 10000 15000 20000 25000 30000
P
R
A
L
S
s
l
Fig. 4. Resul s o he quo um sensing model wi h pa ame e s kRA =.1usingODE
app oach (le ) and me abolic algo i hm ( igh )
3 Towa ds a De ailed P Sys ems Model
Al hough he p elimina y P sys em model desc ibed in Subsec ion 2.2 showed
ha we can ob ain compa able esul s wi h he cu en models p esen ed so
Fig. 8. The a ge Michaelis-Men en concen a ions and he e ol ed P sys ems ones
5 Conclusions and Fu he Wo k
We ha e b iefly desc ibed a pa o he quo um sensing ne wo k in he Pseudomo-
nas ae uginosa. S a ing om a diffe en ial equa ions based model we ha e
p o ided a P sys ems e sion o i and we compa ed he dynamics o he wo
app oaches. In o de o apply diffe en simula ion s a egies on his in iguing
phenomenon we p o ided a mo e de ailed, mechanis ic model which, we belie e,
is close o he biological eali y. The lack o biological in o ma ion ega ding
he dynamics o he sys em led us o use an au oma ic way o es ima ing hem
by using an e olu iona y algo i hm app oach ha offe s a eliable and effec i e
me hod in his espec .
An immedia e s ep u he , a e ob aining all he pa ame e s egula ing a
single bac e ium dynamics, is o ex end he p oposed model a a colony le el,

exploi ing he compa men aliza ion offe ed by P sys ems and al eady es ab-
lished popula ion P sys ems models.
O he impo an de elopmen s a e ela ed o he use o expe imen al da a
o une he dynamics o ou specifica ions such as o simula e eal biological
p ocesses. In his espec he use o model checking me hodologies, al eady unde
conside a ion in a pape unde p epa a ion, will con ibu e owa ds alida ing
ce ain p ope ies o he sys ems modeled.
On long e m we belie e ha hese s eps can ep esen he fi s s age owa d a
quan i a i e analysis ha will hope ully lead o a success ul d ug design p ocess.
Acknowledgemen s. N. K asnogo and P. Siepmann acknowledge he EPSRC
o unding p ojec EP/D021847/1.
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