A S udy o he Robus ness o he EGFR
Signalling Cascade Using Con inuous
Memb ane Sys ems
M.J. P´e ez-Jim´enez and F.J. Rome o-Campe o
Resea ch G oup on Na u al Compu ing,
Depa men o Compu e Science and A i icial In elligence,
Uni e si y o Se illa,
A da. Reina Me cedes s/n, 41012, Se illa, Spain
{ma pe , an}@us.es
Abs ac . Many app oaches o an icance ea men ha e had a limi ed
success. A undamen al hu dle o cance he apy is he obus ness o he
signalling ne wo ks in ol ed in umou genesis. The complexi y o ne -
wo ks o biological signalling pa hways is such ha he de elopmen o
simpli ying models is essen ial in ying o unde s and he wide- anging
cellula esponses hey can gene a e. In his pape a model o he epide -
mal g ow h ac o ecep o signalling cascade is de eloped using con in-
uous memb ane sys ems. This model is used o s udy he obus ness o
his signalling cascade which is known o play a key ole in umou cell
p oli e a ion, angiogenesis and me as asis.
Keywo ds: memb ane compu ing, EGFR signalling ne wo k, signal
ansduc ion, obus ness.
1 In oduc ion
Memb ane Compu ing is an eme gen b anch o Na u al Compu ing in oduced
by Gh. P˘aun in [9]. Since hen i has ecei ed impo an a en ion om he
scien i ic communi y. In ac in 2003 he Ins i u e o Scien i ic In o ma ion (ISI)
has conside ed he seminal pape [9] as as b eaking and Memb ane Compu ing
has been selec ed as a as eme ging a ea in compu e science.
This new non-de e minis ic 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 in a synch onous non-de e minis ic maximally pa allel manne .
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 being able o sol e compu a ionally ha d p oblems in polynomial
ime ading ime o space. P sys ems as a disc e e model o compu a ion ha e
also been used o model biological phenomena (see he olume [1]), and as a
con inuous model in [7]. A i s o maliza ion o non-disc e e P sys em and a
way o app oxima e hem was in oduced in [2].
In his pape we use a con inuous a ian o P sys ems, di e en om ha
in [7], o model he epide mal g ow h ac o ecep o (EGFR) signalling cas-
cade. Up o now he usual ma hema ical o maliza ion o biochemical signalling
ne wo ks has been done using di e en ial equa ions which a e ocused on he
desc ip ion o he change in concen a ion o chemical compounds. He e we use
a new o maliza ion o hese phenomena in a compu a ional amewo k which
ocuses on he compa men al s uc u e (memb ane s uc u e) o he cell and on
he chemical eac ions ( ules) ha ake place in di e en egions o he cell. Thus
his new app oach makes possible a opological and modula modelling o in a-
cellula signalling ne wo ks. In his amewo k expansion o an exis ing model is
done by adding new ules ( eac ions) and, i i is necessa y, new memb anes o
ep esen new egions (o ganella) o he cell; he e o e he p e ious model does
no need o be changed. Besides modula i y and easy ex ensibili y, in a ou o
ou app oach we also men ion he easy unde s andabili y and p og ammabili y,
ea u es which a e no easily achie ed in models which use di e en ial equa ions.
The epide mal g ow h ac o ecep o (EGFR) belongs o he y osine kinase
amily o ecep o s. Binding o he epide mal g ow h ac o (EGF) o he ex a-
cellula domain o EGFR induces ecep o dime iza ion and au ophospho yla ion
o in acellula domains. Then a mul i ude o p o eins a e ec ui ed s a ing a
complex signalling cascade and he ecep o ollows a p ocess o in e naliza ion
and deg ada ion in endosomals. Two p incipal pa hways lead o ac i a ion o
Ras-GTP by hyd oliza ion o Ras-GDP. One o hese pa hways depends on he
concen a ion o he S c homology and collagen domain p o ein (Shc) and he
o he one is Shc-independen . Ras-GTP ac s like a swi ch ha s imula es he Mi-
ogen Ac i a ed P o ein (MAP) kinase cascade by phospho yla ing he p o eins
Ra , MEK and ERK. Subsequen ly phospho yla ed ERK egula es se e al cel-
lula p o eins and nuclea ansc ip ion ac o s. Dis egula ed EGFR exp ession,
ligand p oduc ion and signalling ha e been p o ed o ha e a s ong associa ion
wi h umou genesis. As a esul o his, EGFR has been iden i ied as a key
biological a ge o he de elopmen o no el an icance he apies.
The pape is o ganised as ollows. Con inuous P sys ems a e in oduced in
he nex sec ion. In sec ion 3 he EGFR signalling cascade is b ie ly desc ibed.
We p esen ou model using con inuous P sys ems in sec ion 4. Resul s and
discussion a e exposed in sec ion 5. Finally, conclusions and u u e wo k a e
gi en in he las sec ion.
2 Con inuous P Sys ems
Usual a ian s o P sys ems a e disc e e models o compu a ion whe e in e e y
s ep he ules a e applied in a maximal way an in ege numbe o imes, we
e e o [10] o de ails. He e we use a a ian whose sys ems can e ol e in
e e y ins an applying a maximal se o ules a posi i e eal numbe o imes
de e mined by a ce ain unc ion K. This a ian is inspi ed by he ac ha in
i o chemical eac ions e ol e in a con inuous way ollowing a a e ha depends
on he concen a ion o he eac an s.
Roughly speaking, a con inuous P sys em consis s o a memb ane s uc u e,
a hie a chically a anged se o memb anes, whe e one places mul ise s o ob-
jec s ha ep esen he concen a ion o chemical subs ances. Usual P sys ems
deal wi h disc e e mul ise s o e an alphabe Σbu he e we wo k wi h con inu-
ous mul ise s (mappings om Σ o R+, he se o non-nega i e eal numbe s).
These mul ise s e ol e acco ding o a ini e se o ules ha ep esen chemical
eac ions.
Nex we gi e a o mal de ini ion o con inuous P sys ems. A con inuous P
sys em is a cons uc , Π= (Σ, µ, w1, . . . , wn,R,K), whe e:
1. n≥1 is he deg ee o he sys em (numbe o memb anes).
2. Σ={c1, . . . , cm}is he alphabe o objec s.
3. µis a memb ane s uc u e (a oo ed ee) consis ing o nmemb anes (nodes
o he ee) labelled wi h 1, . . . , n (o en, we iden i y he memb anes wi h
labels om a ini e se H).
4. w1, . . . , wna e con inuous mul ise s associa ed wi h each memb ane o he
memb ane s uc u e µ.
5. Ris a ini e se o ules o he o m:
u[ ]i→u0[ 0]i,
whe e u, ∈Σ∗ ep esen he eac an s, u0, 0∈Σ∗ ep esen he p oduc s,
and i∈His he label o he ele an memb ane o he eac ion ha is
modelled.
6. Kis he a e o applica ion unc ion which associa es wi h each ule and
mul iplici y o he objec s in µ, a non-nega i e eal numbe conside ed as
he a e o applica ion o he ule:
K:R×Mn×m(R+)→R+,
whe e Mn×m(R+) is he se o ma ices o o de n×mo e R+.
Acon igu a ion o a con inuous P sys em Πis a ma ix o Mn×m(R+) whe e
he objec in ow iand column j,ai,j, ep esen s he mul iplici y o he objec cj
in he memb ane i. We in e p e he con igu a ions as assignmen s o con inuous
mul ise s o he memb anes o he sys em, ha is, he associa ion o each egion
wi h he concen a ion o chemical subs ances p esen in i .
Fo usual P sys ems we alk abou compu a ions bu o con inuous P sys ems
we p e e o hink o e olu ions. An e olu ion o a con inuous P sys em is a
mapping om R+ o Mn×m(R+). Tha is, an e olu ion Eassocia es wi h each
ins an ∈R+an ins an aneous con igu a ion E( ) o he sys em:
E( ) = (ai,j( )) 1≤i≤n
1≤j≤m
Fo each ∈R+and i, 1 ≤i≤n, we deno e by i( ) he con inuous mul ise s
o e Σ={c1, . . . , cm}de ined as ollows: ( i( ))(cj) = aij( ) o 1 ≤j≤m. Tha
is, we can desc ibe E( ) by a uple ( 1( ), . . . , n( )).
The way a con inuous P sys em, Π= (Σ, µ, w1, . . . , wn,R,K), e ol es is de-
e mined by he ini ial mul ise s w1, . . . , wnand he a e o applica ion unc ion
K. We de ine he ini ial con igu a ion o Πas he uple (w1, . . . , wn).
The ules a e applied du ing he e olu ion o he sys em in a con inuous way
acco ding o he a e o applica ion unc ion K. A an ins an ∈R+, a ule
∈ R is applied exac ly K( , E( )) imes (in his sense, we can say ha he
ules a e applied in a K-maximal way); ha is, K( , E( )) uni s o he eac an s
a e consumed and K( , E( )) uni s o he p oduc s a e p oduced. Obse e ha
he e ec o he ule dec eases he mul iplici y (concen a ion) o i s eac an s
and inc eases he mul iplici y (concen a ion) o i s p oduc s. Mo e p ecisely, we
de ine he e ec o a ule du ing an in e al o ime [ , T ] as ollows:
E ( , , T ) = ZT
K( , E(s)) ds.
Mo e o mally, gi en an objec cj∈Σ, 1 ≤j≤m, and a memb ane i,
1≤i≤n, we deno e by p oduc ioni(cj) ( esp. consump ioni(cj)) he se o
ules whe e cjis a p oduc in memb ane i( esp. a eac an ). The e o e he eal
numbe ( i( ))(cj), deno ed by |cj|i( ), is de e mined by he nex o mula:
|cj|i( ) = |cj|i(0) + X
∈p oduc ioni(cj)Z
0
K( , E(s)) ds −
−X
∈consump ioni(cj)Z
0
K( , E(s)) ds,
wi h i(0) = wi, ha is, |cj|i(0) = wi(cj)).
In compu e s, eal numbe s a e ep esen ed by a ini e se o a ional numbe s.
The e o e, like in mos con inuous models we need o de elop app oxima ions in
o de o simula e e olu ions o con inuous P sys ems in compu e s.
As shown abo e, in o de o de e mine he e ec o a ule on he e olu ion o
a sys em du ing an in e al o ime [ , T ] we only need o compu e an in eg al o
he a e o applica ion unc ion K. Hence, in o de o app oxima e he e olu ion
o a con inuous P sys ems in a ini e se o ins an s 0,· · · , qwe can use any
sui able known nume ical me hod o app oxima e in eg als. He e o simplici y
we use he ec angle ule; ha is, we suppose l+1 − l=pis small enough o
assume ha K emains cons an and equal o K( , E( l)) in he in e al [ l, l+1]
o l= 0, . . . , q −1. Wi h his assump ion we can app oxima e he e ec o a
ule du ing an in e al o ime o leng h pby E ( , l, l+1)≈pK( , E( l)).
By doing his app oxima ion we each an usual P sys ems ha pe o ms
qs eps ( 0, . . . , q) and in each s eps he ules a e applied pK( , E( l)) imes.
The e o e, we ha e app oxima ed he e olu ion o a con inuous P sys em by he
compu a ion o an usual disc e e P sys em wo king in a pKbounded pa allel
manne .
3 EGFR Signalling Cascade
In his sec ion we will desc ibe b ie ly he EGFR signalling cascade ollowing
he ne wo k depic ed below. Du ing he signal ansduc ion which akes place
in his cascade, he in o ma ion abou he concen a ion o he EGF in he
ou side o he cell is ansla ed in o kine ic in o ma ion inside he cell by EGFR
phospho yla ion.
The epide mal g ow h ac o ecep o (EGFR) belongs o he y osine kinase
amily o ecep o s. The binding o he epide mal g ow h ac o (EGF) o he

ex acellula domain o EGFR induces ecep o dime iza ion and au ophospho-
yla ion o in acellula domains. Then, on he one hand, a mul i ude o p o eins
a e ec ui ed s a ing a complex signalling cascade and, on he o he hand, he
ecep o ollows a p ocess o in e naliza ion, ubiqui ina ion and deg ada ion.
In ou model we conside wo ma ginal pa hways and wo p incipal pa hways
s a ing om he phospho yla ed ecep o .
In he i s ma ginal pa hway phospholipase C-γ(PLCγ) binds o he phos-
pholy a ed ecep o , hen i is phospho yla ed (PLC∗
γ) and eleased in o he cy-
oplasm whe e i can be ansloca ed o he cell memb ane o desphospho yla ed.
In he second ma ginal pa hway he p o ein PI3K binds o he phospholy a ed
ecep o , hen i is phospho yla ed (PI3K∗) and eleased in o he cy oplasm
whe e i egula es se e al p o eins ha we do no include in ou model.
Bo h p incipal pa hways lead o ac i a ion o Ras-GTP. The i s pa hway
does no depend on he concen a ion o he S c homology and collagen domain
p o ein (Shc). This pa hway consis o a cycle whe e he p o eins g ow h ac o
ecep o -binding p o ein 2 (G b2) and Son o Se enless homolog p o ein (SOS)
bind o he phospho yla ed ecep o . La e he complex G b2-SOS is eleased in
he cy oplasm whe e i dissocia es in o G b2 and SOS.
In he o he main pa hway Shc plays a key ole, i binds o he ecep o
and i is phospho yla ed. Then ei he Shc∗is eleased in he cy oplasm o he
p o eins G b2 and SOS binds o he ecep o yielding a ou p o ein complex
(EGFR-EGF2*-Shc*-G b2-SOS). Subsequen ly his complex dissocia es in o he
complexes Shc∗-G b2-SOS, Shc∗-G b2 and G b2-SOS which in u n can also
dissocia e o p oduce he p o eins Shc∗, G b2 and SOS.
Finally, Ras-GTP is ac i a ed by hese wo pa hways and in u n i s imu-
la es he Mi ogen Ac i a ed P o ein (MAP) kinase cascade by phospho yla ing
he p o eins Ra , MEK and ERK. Subsequen ly phospho yla ed ERK egula es
se e al cellula p o eins and nuclea ansc ip ion ac o s ha we do no include
in ou model.
The e exis c oss- alks be ween di e en pa s and cycles o he signalling
cascade which sugges a s ong obus ness o he sys em.
Fo a mo e de ailed desc ip ion o he cascade see he li e a u e lis ed in he
bibliog aphy.
4 Modelling EGFR Signalling Cascade by Con inuous P
Sys ems
We ha e de eloped a model o he signalling cascade desc ibed in he p e i-
ous sec ion using a con inuous P sys em, ΠEGF = (Σ, µ, we, ws, wc,R,K). Ou
model consis s o mo e ha 60 p o eins and complexes o p o eins and 160
chemical eac ions. Supplemen a y in o ma ion and de ails abou he model a e
a ailable on he web page www.gcn.us.es/eg .pd .
•Alphabe : In he alphabe Σwe collec all he p o eins and complexes o
p o eins ha ake pa in he signalling cascade. In able 1 o he Supplemen a y
in o ma ion all he objec s o he alphabe and he chemical compounds ha
hey ep esen a e lis ed.
•Memb ane S uc u e: In he EGFR signalling cascade desc ibed in he
p e ious sec ion, he e a e h ee ele an egions, namely he en i onmen , he
cell su ace and he cy oplasm. We ep esen hem in he memb ane s uc u e as
he memb anes labelled wi h: e o he en i onmen , s o he cell su ace and c
o he cy oplasm.
•Ini ial Mul ise s: In he ini ial mul ise s we ep esen he ini ial concen a-
ions o he chemical subs ances in he en i onmen , he cell su ace and he
cy oplasm. These concen a ion has been ob ained om he e e ences lis ed in
he bibliog aphy. A de ailed p esen a ion o ini ial mul ise s is shown in able 2
o he Supplemen a y in o ma ion.
•Rules and Ra e o applica ion unc ion: In he ules we model he chemi-
cal eac ions desc ibed which o m he signalling cascade. To model he eac ions
we use he Law o Mass Ac ion which s a es ha he a e o a eac ion is p o-
po ional o he p oduc o he concen a ions o he eac an s. Tha is, i we
ha e a eac ion o he o m:
1+· · · + k→p1+· · · +pk0,
hen he a e o his eac ion is k| 1| · · · | n|, whe e kis called kine ic cons an .
In ables 3-11 o he Supplemen a y in o ma ion (www.gcn.us.es/eg .pd )
all he ules a e lis ed as well as he kine ic cons an s and he e e ences om
whe e hey we e aken. As an example o he p ocedu e we ha e ollowed o
de elop ou model, we nex p esen he de i a ion o one o he 160 ules.
Le us conside he binding o EGF o EGFR:
EGF EGFR →EGF-EGFR
We know om biological expe imen s ha EGF, which is p esen in he
en i onmen , binds o EGFR, which is p esen in he cell su ace a a a e o
0.003 nM−1s−1. Acco ding o his, he ele an memb ane in his eac ion is
he cell-su ace because i sepa a es he wo egions in ol ed in his eac ion.
Besides ollowing he Mass Ac ion Law he eac ion akes place a a eloci y o
0.003|EGF ||EGF R|. The e o e, in ou model we ep esen his chemical eac ion
by he ollowing ule and a e o applica ion:
EGF [EGFR]s→[EGF-EGFR]sK( , E( )) = 0.003|EGF ( )|e|EGF R( )|s
5 Resul s and Discussion
The model p esen ed in he p e ious sec ion has been implemen ing using CLIPS,
a p oduc i e de elopmen and deli e y expe sys em ool which p o ides a
comple e en i onmen o he cons uc ion o ule and/o objec based expe
sys ems.
To implemen ou model we ha e app oxima ed he e olu ions o he con-
inuous P sys em ΠEGF by compu a ions o an usual P sys em wo king in a
bounded pa allel manne . The pa ame e p, chosen o he app oxima ion, was
ixed o 10−3a e es ing di e en alues un il he esul s ob ained did no
change.
We s udy he e ec o di e en EGF concen a ions on he signalling cascade.
To illus a e his e ec we depic he e olu ion o he concen a ion o he mos
ele an p o eins in he signalling cascade o e ime.
010 20 30 40 50 60
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
ime (s)
Concen a ion (nM)
100nM
200nM
300nM
Abo e i can be seen ha he ecep o ac i a ion by au ophopho yla ion is
clea ly concen a ion dependen showing a high peak in he i s 5 seconds o
decay apidly a e wa ds o e y low le els o concen a ion. Acco ding o he
a iance in he ecep o ac i a ion i is in ui i e o expec di e en cell esponses
o di e en EGF concen a ions. He e we will show ha his is no he case. Nex
we show he e olu ion o he phospho yla ion o Shc a e binding o he ecep o
o e 60 seconds.
010 20 30 40 50 60
0
2
4
6
8
10
12
14
16
ime (s)
Concen a ion (nM)
100nM
200nM
300nM
I can be obse ed ha he esponses o EGF s imula ion ge mo e sus ained
as we ge deepe in he cascade and ha he ne wo k has almos managed o
a enua e he o e s imula ion wi h EGF. Obse e ha he ed and blue line a e
almos iden ical.
The ac i a ion o he Ras p o ein is a key node in he EGFR signalling
cascade. We depic i s e olu ion o e 180 seconds in he nex g aphic.
020 40 60 80 100 120 140 160 180
0
10
20
30
40
50
ime (s)
Concen a ion (nM)
100nM
200nM
300nM
No e ha a his poin he esponse o he o e s imula ion wi h EGF has
been comple ely a enua ed and ha an ampli ica ion o he esponse o he low
concen a ion has been pe o med.
Finally, he goal o he cascade is he ac i a ion o he mi ogenic kinases MEK
and ERK. These p o eins egula es he ansc ip ion o se e al p o o-oncogenes
like c- os and c-Jun. Nex he e olu ion o phospho yla ed MEK o e 180 second
is p esen ed.
020 40 60 80 100 120 140 160 180
0
1
2
3
4
5
ime (s)
Concen a ion (nM)
100nM
200nM
300nM
In his pic u e i is shown he su p ising obus ness o he signalling cascade.
The signals om ou side due o EGF concen a ion ha e been ei he a enua ed
o ampli ied o ge he same concen a ion o he mos ele an kinases. Obse e
ha a e 100 seconds, when he esponse ge s sus ained, he h ee line ep esen -
ing he esponse o di e en ex e nal EGF concen a ions a e iden ical. Fo mo e