How o Exp ess Tumou s Using
Memb ane Sys ems
Miguel A. Gu i´e ez-Na anjo, Ma io J. P´e ez-Jim´enez,
Agus ´ın Riscos-N´u˜nez, F ancisco J. Rome o-Campe o
Dp o. de Ciencias de la Compu aci´on e In eligencia A i icial
E.T.S. Ingenie ´ıa In o m´a ica, Uni e sidad de Se illa
A da. Reina Me cedes s/n, 41012, Se illa, Espa˜na
{magu ie , ma pe , a iscosn, an}@us.es
Janua y 12, 2007
Abs ac
In his pape we discuss abou he po en ial use ulness o memb ane
sys ems as na u al ools o modelling umou s. This is done bo h om
a mac oscopic and a mic oscopic poin o iew. In he i s case, one
conside s he umou as a g owing mass o cells, ocusing on i s ex e nal
shape. In he second case, one descends o he mic oscopic le el, s udying
molecula signalling pa hways ha a e c ucial o de e mine i a cell is
cance ous o no . In each o hese app oaches we wo k wi h app op ia e
a ian s o memb ane sys ems.
1 In oduc ion
Na u al Compu ing is a ield ha ies o abs ac ideas and new pa adigms
o in oducing models o compu a ion inspi ed by Na u e. One o he b anches
wi hin his ield is Memb ane Compu ing, p esen ed by Gh. P˘aun in [15], whe e
he basic compu ing de ices a e he so-called memb ane sys ems o P sys ems1.
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 (associa ed wi h memb anes) in a synch onous non-
de e minis ic maximally pa allel manne .
F om a classical compu e science poin o iew, basic P sys ems can be
conside ed as collec ions o pa allel p ocesso s placed in a ee s uc u e (each
p ocesso plays he ole o a memb ane). Ac ually, he e exis wo le els o
pa allelism in he model: on he one hand, he ules wi hin a memb ane a e
1A layman-o ien ed in oduc ion can be ound in [17], a o mal desc ip ion in [16], and
u he up- o-da e bibliog aphy a [25].
1
applied simul aneously; on he o he hand, hese ope a ions a e pe o med in
pa allel in all he memb anes o he sys em.
In his pape we add ess he issue o using P sys ems in o de o p o ide a
be e unde s anding o umou s. Fi s , we will conside a mac oscopical poin
o iew, looking a he whole umou . In pa icula , we will ollow a numbe o
ecen s udies (see [1, 3, 5, 9, 10, 12, 20, 21]) pos ula ing ha umou s ha e a
ac al shape, and we show how o use P sys ems as a new ool o ep esen ing
and simula ing such a ac al na u e o umou s. The choice o P sys ems is
based on hei massi e pa allelism, he synch onous applica ion o hei ules,
and he disc e e na u e o hei compu a ion, among o he ea u es.
F om a mesoscopic poin o iew (i.e. a a cellula le el), one can y o un-
de s and, by means o P sys ems, some biosignalling ne wo k unc ions ha a e
e y impo an o s udying di e en diseases. In pa icula , one can s udy some
p o einic in e ac ions happening om he ex acellula domain o he cy oplasm
o umo al cells. We will p esen some ideas om [19] conce ning he epide mal
g ow h ac o ecep o (EGFR) signalling pa hway ha pe mi o jus i y he
obus ness o he EGFR sys em (cha ac e ised by he simple beha iou ha
a ises om complex p ocesses).
The pape is o ganised as ollows: Fi s we deal wi h umou s a a mac o-
scopic le el. We ecall some ideas abou ac als and he de ini ion o P sys ems
wi h memb ane c ea ion in Subsec ion 2.2. Nex , we explain how he e olu ion
o a P sys em can be linked o he cons uc ion o a classical ac al, he Koch
cu e. Subsec ion 2.4 concludes he mac oscopic app oach, add essing andom
ac als, which a e close o he eal shape o umou s. Sec ion 3 is de o ed o
he in acellula scena io o umo al cells. In his case we ecall he de ini ions o
P sys ems used o model signalling pa hways and we b ie ly p esen he EGFR
signalling pa hway in Subsec ion 3.2. We conclude his pape by discussing
some inal ema ks abou he link ha has been es ablished be ween memb ane
compu ing and cance esea ch.
2 Mac oscopic View: Tumou s and P Sys ems
An indi idual umo al cell has he po en ial, o e successi e di isions, o de elop
in o a clus e o umo al cells. Fu he g ow and p oli e a ion leads o he
de elopmen o an a ascula umou consis ing o app oxima ely 106cells which
eed on oxygen and o he nu ien s p esen in he local en i onmen .
The apid g ow h and esilience o umou s make i di icul o belie e ha
hey beha e as diso ganised and di use cell mass and sugges s ins ead ha
hey a e eme ging, oppo unis ic sys ems. I his hypo hesis holds ue, hen
s udying single cells is no enough, he g owing umou as a whole mus be
in es iga ed and ea ed as a sel -o ganising complex dynamic sys em.
A i s app oach o he simula ion o a umou g ow h by using P sys ems
was p oposed in [7]. The model ollowed he e was he sphe oid model, whe e
he umou is conside ed as a mul icell sphe oid wi h a cha ac e is ic s uc u e
o a p oli e a ing im and a nec o ic co e, sepa a ed by a band o quiescen cells.
2
The sphe oid model has e y nice ma hema ical p ope ies, bu ecen s ud-
ies in he g ow h o umou s show ha i is oo simple in o de o explain he
g ow h o he umou in a ealis ic way. Indeed, he su ace o he umou is
a om being a smoo h su ace, and ac ually i seems ha his is one o he
cases in which Na u e p oduces a ac al-like su ace. E en mo e, he e seems
o be a ela ion be ween he ac al dimension o he su ace o he umou and
he s ages o he disease. In [10], Kikuchi e al. poin ha he su ace o solid
componen s in cys ic epi helial o a ian cance s has a ac al s uc u e and he
mean ac al dimension may di e acco ding o he s ages o he disease and his-
ologic ypes. F ac al geome y (. . . ) can be used o desc ibing he pa hological
a chi ec u e o o a ian umou s and o yielding insigh s in o he mechanisms
o umou g ow h.
These s udies show he necessi y o going deepe in he ela ion be ween
ac als and umou s, which is cu en ly maybe one o he mos p omising ap-
plica ions o ac als. This s udy will need new ools o handling in o ma ion
and o compu ing/simula ing/p edic ing esul s. As poin ed by Baish and Jain
in [1]: I ca e ully applied, ac al me hods may someday ha e a signi ican im-
pac in ou unde s anding o challenges in ea men deli e y and diagnosis o
cance .
2.1 F ac als
One o he mos impo an ea u es o ac als is ha hey a e a om being
me ely ma hema ical cu iosi ies o compu a ional a objec s. F ac als a e one
o he mos powe ul ools o desc ibing many na u al objec s bo h om ali e
and non-ali e wo ld2.
The seminal wo k on ac als, p esen ed by Mandelb o [13] in 1982, pu he
basis o he heo y dealing wi h ma hema ical se s ha a e no egula enough.
In a i s app oach, we can conside ha ac al objec s exhibi complexi y which
holds cons an unde di e en scales. A ac al is a shape made o pa s simila
o he whole, in some way.
Nowadays he e is no de ini ion o ac als which conside s e e y case. In-
s ead o a o mal de ini ion, a se Fis conside ed a ac al (in an in o mal sense)
i i ul ils se e al p ope ies (see [6] and [23]):
•Fhas a ine s uc u e, i.e., de ail on a bi a y many scales.
•Fis oo i egula o be desc ibed in adi ional geome ical language, bo h
locally and globally.
•O en Fhas some o m o sel -simila i y.
•Usually, i s ac al dimension (de ined in some way) is g ea e han i s
opological dimension.
•F ac als a e ob ained by he applica ion o ecu si e p ocedu es, usually
in a simple way. These p ocedu es o en consis o ew ules.
2Fo mo e applica ions o ac als, see, o example, [4, 18, 23].
3
•The compu a ional gene a ion o a ac al is disc e e. F ac als a e usually
de ined as he limi o an i e a i e p ocess pe o med s ep by s ep.
Sel -simila i y seems o be one o he undamen al geome ical cons uc ion
p inciples in Na u e. In many plan s and also in animal o gans, his has led
o ac al b anching s uc u es. Fo example, he b anching s uc u e in a ee
allows he cap u e o a maximum amoun o sun ligh by he lea es; he blood
essel sys em in a lung is simila ly b anched so ha he maximum amoun
o oxygen can be assimila ed (see [18]). Al hough he sel -simila i y in hese
objec s is no s ic , we can iden i y he building blocks o he s uc u e.
Nex , we p esen a a ian o P sys ems sui able o b idge Memb ane Com-
pu ing and F ac als, namely P sys ems wi h memb ane c ea ion.
2.2 P Sys ems wi h Memb ane C ea ion
Memb anes a e c ea ed in li ing cells, o ins ance, in he p ocess o esicle
media ed anspo , and also in o de o keep molecules close o each o he
acili a ing hei eac ions. One can abs ac he biological ope a ion o c ea ing
new memb anes unde he in luence o exis ing chemical subs ances o de ine P
sys ems wi h memb ane c ea ion.
Recall ha a P sys em wi h memb ane c ea ion o ini ial deg ee m≥1 is a
uple o he o m Π = (O, H, µ, w1,...,wm, R) whe e:
1. Ois he alphabe o objec s;
2. His a ini e se o labels o memb anes;
3. µis a memb ane s uc u e consis ing o mmemb anes labelled (no nec-
essa ily in a one- o-one manne ) by elemen s o H;
4. w1,...,wma e s ings o e O, desc ibing he mul ise s o objec s placed
in he m egions o µ;
5. Ris a ini e se o ules, o he ollowing o ms3:
(a) [a→ ]h, whe e h∈H,a∈O, and is a s ing o e Odesc ibing a
mul ise o objec s. These a e objec e olu ion ules associa ed wi h
memb anes and depending only on he label o he memb ane.
(b) a[ ]h→[b]h, whe e h∈H,a, b ∈O. These a e send-in commu-
nica ion ules. An objec is in oduced in he memb ane, possibly
modi ied.
(c) [a]h→[ ]hb, whe e h∈H,a, b ∈O. These a e send-ou communica-
ion ules. An objec is sen ou o he memb ane, possibly modi ied.
3In his pape we will use a weak e sion o his model, since we do no use dissolu ion no
communica ion ules, none heless we p esen he model o P sys ems wi h memb ane c ea ion
as ound in he li e a u e.
4
(d) [a]h→b, whe e h∈H,a, b ∈O. These a e dissolu ion ules. In
eac ion wi h an objec , a memb ane is dissol ed, while he objec
speci ied in he ule can be modi ied.
(e) [a→[ ]h2]h1, whe e h1, h2∈H,a∈O, and is a s ing o e O
desc ibing a mul ise o objec s. These a e c ea ion ules. In eac ion
wi h an objec , a new memb ane is c ea ed. This new memb ane,
labelled by h2, is placed inside he memb ane whe e he objec which
igge s he ule was loca ed, and has associa ed wi h i an ini ial
mul ise .
Rules a e applied acco ding o usual p inciples in Memb ane Compu ing (see
[8] o de ails).
2.3 A Memb ane Sys em Building he Koch Cu e
Le us jus p esen he e one o he classic ac als, he Koch cu e [11], and
a P sys em which can be “in e p e ed” as his ac al, in a sense ha will be
discussed below. Following Ba nsley [2], in o de o desc ibe he ac al we need
o know he ini ial condi ions (o ini ial con igu a ion in e ms o memb ane
compu ing) and he ans o ma ion ules, bu we also need a u he ing edien :
as in e e y compu a ional p ocess, we s o e he in o ma ion in some kind o
da a s uc u e and in o de o ecognise he da a as a ac al we need o gi e an
in e p e a ion o he da a.
The geome ic cons uc ion o he Koch cu e can be easily desc ibed. Le
us begin wi h a s aigh segmen K0which we will conside o leng h one, hen
we spli K0in o h ee segmen s o leng h 1/3 and we eplace he middle hi d
by an equila e al iangle and ake away i s base.
The e o e, he nex s age on he cons uc ion o he Koch cu e, K1consis s
o a con inuous line composed by ou s aigh segmen s o leng h 1/3. The
cons uc ion con inues by i e a ing his p ocedu e.
I we ollow a simila cons uc ion s a ing om an equila e al iangle in-
s ead o s a ing om a segmen , hen we ob ain he so-called Koch snow lake,
depic ed in Figu e 1.
Nex , we p esen a P sys em which can be in e p e ed as he Koch cu e.
In each s ep o compu a ion we ha e a con igu a ion co esponding o an in-
e media e s ep o he cons uc ion o he Koch cu e. Le us conside he P
Figu e 1: Fi s s eps o he Koch Snow lake
5

sys em Π = (O, H, µ, w4, R) wi h O={a, b, c, α, β, γ},H={1,2,3,4},µ= [ ]4,
w0={abcγ}, and R he ollowing se o 16 ules
R1= [ α→[abcα ]4]1,R2= [ β→[abcβ ]4]2,
R3= [ α→[abcα ]4]3,R4= [ γ→[abcγ ]4]4,
Ri
a= [ a→[abcα]1]i, i ∈ {1,2,3,4},
Ri
b= [ b→[abcβ]2]i, i ∈ {1,2,3,4},
Ri
c= [ c→[abcα]3]i, i ∈ {1,2,3,4}.
In o de o in e p e each con igu a ion, we collec o each elemen a y mem-
b ane he s ing composed by all he labels o he in e media e memb anes
be ween he conside ed elemen a y memb ane and he skin, and we ake in o
accoun he dep h (in he memb ane s uc u e) o he elemen a y memb anes,
as well as he symbol occu ing in he memb ane deno ed by a G eek le e
(which can be ei he α, β, o γ).
•Each elemen a y memb ane will ep esen a segmen .
•The dep h o he elemen a y memb ane in he memb ane s uc u e will
de e mine he leng h o he segmen ep esen ed by he memb ane. We
will conside ha he skin has dep h 0 and ha an elemen a y memb ane
a dep h k ep esen s a segmen o leng h 1/3k.
•We will use he s ing o labels o he in e media e memb anes om he
elemen a y memb ane o he skin o “o de ” he elemen a y memb anes
ollowing he o de <Sde ined as ollows. Le us conside w1, w2∈H∗
such ha w1is no a su ix o w2and ice e sa (consequen ly, w16=w2),
hen we will say ha w1<Sw2i and only i he e exis z1, z2, w ∈H∗
and x1, x2∈Hwi h w1=z1x2w,w2=z2x2wand x1< x2(whe e <is an
o de o e H). No e ha <Sis a so o lexicog aphic o de , bu s a ing
om igh o le .
•Each s age o he cons uc ion o he Koch cu e consis s on a con inuous
line buil wi h a ce ain amoun o segmen s, all o hem wi h he same
leng h. I we know he numbe o such segmen s and hei leng h, hen
in o de o de e mine exac ly an in e media e s ep o he cons uc ion o
he Koch cu e, he las da a ha we need is he angle be ween adjacen
segmen s. This in o ma ion is gi en by he symbol α,β, o γplaced inside
each elemen a y memb ane.
–I a memb ane ep esen ing a segmen scon ains he symbol α, hen
we will conside ha he ollowing segmen has a de ia ion o π/3
adians wi h espec o he di ec ion o s.
–I a memb ane ep esen ing a segmen scon ains he symbol β, hen
we will conside ha he ollowing segmen has a de ia ion o −2π/3
adians wi h espec o he di ec ion o s.
6
–Finally, i a memb ane ep esen ing a segmen scon ains he symbol
γ, we will conside ha i is he las segmen o he line and no o he
segmen is a e i .
The ini ial con igu a ion has only one memb ane, he skin, and he objec s
abcγ inside. Wi h he in e p e a ion speci ied abo e, his ini ial con igu a ion
C0 ep esen s a unique segmen o leng h 1/30= 1. The G eek symbol inside is
γand, cohe en ly, i means ha no o he segmen is a e i .
By he applica ion o ules R4
a,R4
b,R4
c, and R4, we ob ain he con igu a ion
C1= [ [abcα]1[abcβ]2[abcα]3[abcγ]4]4.
This con igu a ion has ou elemen a y memb anes a dep h 1 ha ep esen
ou segmen s o leng h 1/31. The o de among he s ings o labels is 14 <S
24 <S34 <S44. The i s segmen (wi h s ing 14) con ains he symbol α.
This means ha he second segmen (wi h s ing 24) has a de ia ion o π/3
wi h espec he di ec ion o he i s one. The second segmen con ains he
symbol β, so we will conside ha he hi d segmen has a de ia ion o −2π/3
wi h espec o he second one. Analogously, he ou h segmen has a de ia ion
o π/3 wi h espec o he he hi d one, since in he hi d memb ane we ind
he symbol α. Finally, in he las memb ane we ind he symbol γma king he
end poin .
Wi h his in e p e a ion, he con igu a ion C1con ains all he necessa y
in o ma ion o cons uc he s age K1 o building he Koch cu e.
A e js eps, we each a con igu a ion wi h 4jelemen a y memb anes a
dep h jwi h a sequence o angles ha co esponds o he s age Kjo he Koch
cu e.
Thus, he abo e P sys em encodes all he in o ma ion needed o build he
Koch cu e wi h any p ecision deg ee.
2.4 Random F ac als and P Sys ems
The Koch snow lake is no pe cei ed as a close model o eal wo ld ac als (e.g.
a coas line o a umou ). The eason lies in he lack o andomness. Randomis-
ing a de e minis ic classical ac al is he i s app oach gene a ing a ealis ic
na u al shape.
Fo example, he me hod o including andomness in he Koch snow lake
cons uc ion equi es only a e y small modi ica ion o he classical cons uc ion.
A s aigh line segmen will be eplaced as be o e by a b oken line o ou
segmen s, each one one- hi d as long as he o iginal segmen . Howe e , he e
a e wo possible o ien a ions in he eplacemen s ep: he small angle may go
ei he o he le o o he igh . I one o hese o ien a ions is chosen in each
eplacemen s ep, we ob ain a andom Koch cu e. Figu e 2 shows a andom
Koch snow lake. No e ha his ac al ep esen s a ealis ic shape o a ac al
om Na u e.
In his p ocess some ma hema ical cha ac e is ics o he Koch snow lake will
be e ained, o example he ac al dimension o he cu e will be he same,
7
he a ea su ounded is ini e bu he leng h o he cu e is in ini e, e c., bu he
isual appea ance is d as ically di e en : i looks much mo e like he ou line o
he island o like a umou han he o iginal Koch cu e.
In he p e ious subsec ion we p esen ed a P sys em encoding he Koch cu e
wi h an app op ia e in e p e a ion o he objec s. I was a de e minis ic sys em,
and in i s compu a ion he con igu a ion Cnwas iden i ied wi h he n- h s age
in he cons uc ion o he ac al.
The cons uc ion o andom ac als wi h P sys ems can be pe o med in
a e y na u al way by using he non-de e minism o P sys ems. Fo example,
in he i s s ep o he cons uc ion o he Koch cu e we s a wi h a s aigh
line and he e a e wo possible new eachable s ages. Analogously, we can
modi y he P sys em p esen ed in subsec ion 2.3 and ob ain a non-de e minis ic
P sys em such ha wo possible con igu a ions a e eachable om he ini ial
con igu a ion. Each o hese con igu a ions can be in e p e ed as one o he
eachable s ages in he cons uc ion o he Koch cu e.
Le us conside he ollowing P sys em wi h ini ial deg ee 1,
Π = (O, H, µ, w4, R),
wi h O={s, a , b , c , al, bl, cl, α , β , αl, βl, γ},H={1,2,3,4},µ= [ ]4,w4=
{sγ}, and R he ollowing se s o ules
[s→a b c ]i
[s→alblcl]ii∈ {1,2,3,4}
[αj→α0
j]1
[βj→β0
j]2
[αj→α0
j]3



j∈ {l, }
[γ→γ0]4
[aj→[sαj]1]i
[bj→[sβj]2]i
[cj→[sαj]3]i



i∈ {1,2,3,4}
j∈ {l, }
[α0
j→[sαj]4]1
[β0
j→[sβj]4]2
[α0
j→[sαj]4]3



j∈ {l, }
[γ0→[sγ ]4]4
Figu e 2: Random Koch Snow lake
8
In his case, he in e p e a ion o he P sys em as a ac al is a li le bi
mo e complica ed. In he same way as in he de e minis ic Koch cu e, we
will conside he elemen a y memb anes, hei dep h, and o each elemen a y
memb ane, he s ing o labels o he memb anes om he elemen a y memb ane
o he skin and he special symbol placed in he memb ane, aken om he se
{α , β , αl, βl, γ}.
The in e p e a ion o hese special symbols is qui e na u al. In he de e -
minis ic case, i a memb ane ep esen ing a segmen scon ains he symbol α, we
conside ed ha he nex segmen had a de ia ion o π/3 adians wi h espec
o he di ec ion o s. In he andom case, α ep esen s a de ia ion o π/3 and
αla de ia ion o −π/3. Analogously, β ep esen s a de ia ion o −2π/3 and
βla de ia ion o 2π/3. Finally, i a memb ane con ains he symbol γ, we will
conside ha i ep esen s he las segmen o he line, and no o he segmen
comes a e i .
The main di e ence consis s on he ac ha in his P sys em only con igu a-
ions in e en s eps will be in e p e ed as in e media e s ages o he cons uc ion
o he ac al. Odd s eps a e auxilia y s eps wi hou geome ical in e p e a ion.
The emaining in o ma ion is s o ed in he same way as in he de e minis ic
case.
3 Mic oscopic View: P Sys ems Modelling Molec-
ula In e ac ions
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 no easily
achie ed in s anda d models used nowadays, mainly based on di e en ial equa-
ions. In his sec ion we p opose P sys ems as a amewo k o modelling can-
ce ous p ocesses, om an in acellula poin o iew. Ins ead o conside ing he
umou as a whole, he e we use P sys ems in o de o simula e (some) signalling
pa hways occu ing inside he cell.
3.1 P Sys ems o model signalling pa hways
In his pape we conside he syn ac ic a ian o P sys ems being uples
Π = (O, L, µ, M1, M2,...,Mn, R1,...,Rn) ,
whe e:
•Ois a ini e alphabe o symbols ep esen ing objec s (p o eins and com-
plexes o p o eins);
•Lis a ini e alphabe o symbols ep esen ing labels o he compa men s
(memb anes);
•µis a memb ane s uc u e con aining n≥1 memb anes labelled wi h
elemen s om L;
9