A Membrane Computing View on Tumours

A Memb ane Compu ing View on Tumou s 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 Abs ac . In his pape we discuss abou he po en ial use ulness o P sys ems as na u al ools o modelling umou s. This is done bo h om a mac oscopic poin o iew, by conside ing he umou as a g owing mass o cells, as well as om a mic oscopic poin o iew, by s udying molecula signalling pa hways. In each o hese app oaches we wo k wi h app op ia e a ian s o P 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 [16], whe e he basic compu ing de ices a e he so-called memb ane sys ems o 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 1. P sys ems o e wo le els o pa allelism: on he one hand, he ules wi hin a memb ane a e 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. To sum up, P sys ems ha e he ollowing p ope ies: –P sys ems can be conside ed as s uc u es o nes ed p ocesso s placed in a ee-s uc u e, i.e., we can conside compu a ions on many scales. –I we conside P sys ems whe e memb anes can be dissol ed, di ided o c ea ed, we usually ob ain a geome ical shape oo i egula o be desc ibed in adi ional geome ical language, bo h locally and globally. –Compu a ions in P sys ems a e ob ained by he applica ion o a ini e se o ules. The applica ion o hese ules allows o ob ain a con igu a ion Cn+1 om ano he con igu a ion Cn. –The compu a ion o a P sys em is disc e e, i.e., i is a p ocess pe o med s ep by s ep. 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. F om an in a-cellula poin o iew, one can y o exp ess he molecula in e ac ions happening in he cy oplasm o a umo al cell by means o P sys ems. We will ecall some ideas om [20], whe e he EGFR signalling pa hway is s udied, bu we a e no going o ocus on his app oach. We will also conside a mac oscopical poin o iew, looking a he whole umou . 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, he g owing umou and no only he single cell mus be in es iga ed and ea ed as a sel -o ganising complex dynamic sys em. This canno be done wi h cu en ly a ailable in i o/in i o models o common ma hema ical app oaches. We ollow he e a numbe o ecen s udies (see [1, 3, 5, 10, 11, 14, 21, 22]) pos ula ing ha umou s ha e a ac al shape. The massi e pa allelism, he synch onous applica ion o he ules, and he disc e e 1A layman-o ien ed in oduc ion can be ound in [18], a o mal desc ip ion in [17], and u he bibliog aphy a [24]. 104 M.A. Gu i´e ez-Na anjo e al. na u e o hei compu a ion, among o he ea u es, lead us o conside P sys ems as na u al ools o dealing wi h ac als. Se e al examples o ac als ep esen ed by P sys ems a e p esen ed, and we p opose o use P sys ems as a new ool o ep esen ing and simula ing he ac al na u e o umou s. The pape is o ganised as ollows. Fi s we deal wi h umou s a mac oscopic le el. We ecall de de ini ion o P sys ems wi h memb ane c ea ion in Subsec ion 2.1, and 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 2.3 concludes he mac oscopic app oach, and i add esses 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 con inuous P sys ems and we b ie ly p esen he EGFR signalling pa hway in Subsec ion 3.2. 2 Mac oscopic View: Tumou s and P Sys ems An indi idual umou cell has he po en ial, o e successi e di isions, o de elop in o a clus e o u- mou 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 . In [8] we p oposed a i s app oach o he simula ion and he s udy o he g ow h o a umou by using P sys ems. The model ollowed he e was he sphe oid model. The as majo i y o classic models apply speci ically o mul icell sphe oids which ha e 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. The sphe oid model has e y nice ma hema ical p ope ies, bu ecen s udies in he g ow h o umou s show ha he su ace o he umou is a om being a smoo h 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 [11], 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, and his is maybe one o he mos p omising applica ions o ac als. This s udy will need new ools o handling in o ma ion and 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 impac in ou unde s anding o challenges in ea men deli e y and diagnosis o cance . Nex , we p esen a a ian o P sys em sui able o b idge Memb ane Compu ing and F ac als, namely P sys ems wi h memb ane c ea ion. 2.1 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 in o de o keep molecules close o each o he o acili a e hei eac ions. He e we abs ac he ope a ion o c ea ion o 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 is a uple o he o m Π= (O, H, µ, w1, . . . , wm, R) whe e: 1. m≥1 is he ini ial deg ee o he sys em; 2. Ois he alphabe o objec s; 3. His a ini e se o labels o memb anes; 4. µis a memb ane s uc u e consis ing o mmemb anes labeled (no necessa ily in a one- o-one manne ) wi h elemen s o H; 5. w1, . . . , wma e s ings o e O, desc ibing he mul ise s o objec s placed in he m egions o µ; 6. Ris a ini e se o ules2, o he ollowing o ms: 2In his pape we will use a weak e sion o his model, since we do no use dissolu ion no communica ion ules, bu we wan o p esen he model o P sys em wi h memb ane c ea ion as ound in he li e a u e. A Memb ane Compu ing View on Tumou s 105 (a) [a→ ]h, whe e h∈H,a∈Oand 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 communica 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. (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∈Oand is a s ing o e Odesc ibing a mul ise o objec s. These a e c ea ion ules. In eac ion wi h an objec , a, a new memb ane is c ea ed. This new memb ane is placed inside o he memb ane o he objec which igge s he ule and has associa ed an ini ial mul ise and a label, h2. Rules a e applied acco ding o usual p inciples in Memb ane Compu ing (see [9] o de ails). 2.2 The Koch Cu e 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 ld3. The seminal wo k on ac als p esen ed by Mandelb o [15] in 1982 pu he basis o he heo y o deal wi h ma hema ical se s no su icien ly egula . 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 a de ini ion o ac als which conside s e e y case. Ins ead o a o mal de - ini ion, a se Fis conside ed a ac al (in in o mal sense) i i ul ils se e al p ope ies (see [7] 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 a ew ules. –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 o gans o animals, his has led o ac al b anching s uc u es. Fo example, in a ee he b anching s uc u e 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 [19]). 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. An in–de ail in oduc ion on ac als alls ou o he scope o his pape . Le us jus p esen he e one o he classic ac als, he Koch cu e [12, 13], 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 o 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 s uc u e da a 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. In a simila way as in he middle hi d Can o se , 3Fo mo e applica ions o ac als, see, o example, [4, 6, 19, 23]. 106 M.A. Gu i´e ez-Na anjo e al. we spli K0in o h ee segmen s o leng h 1/3. Then 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 on leng h 1/3. 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 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,Ri a= [ a→[abcα]1]i, i ∈ {1,2,3,4}, R2= [ β→[abcβ ]4]2,Ri b= [ b→[abcβ]2]i, i ∈ {1,2,3,4}, R3= [ α→[abcα ]4]3,Ri c= [ c→[abcα]3]i, i ∈ {1,2,3,4}, R4= [ γ→[abcγ ]4]4. In each con igu a ion, we collec o each elemen a y memb 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 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 he 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. •Finally, i a memb ane ep esen ing he 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 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 . Fig. 1. Fi s s eps o he Koch Snow lake A Memb ane Compu ing View on Tumou s 107 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 <S24 <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 ound he symbol α. Finally, in he las memb ane we ound he symbol γ o ma k 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 equal 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. 2.3 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. Randomising a de e minis ic classical ac al is he i s app oach gene a ing a ealis ic na u al shape. Fig. 2. Random Koch Snow lake 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, 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 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. 108 M.A. Gu i´e ez-Na anjo e 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 Sec ion 2.2 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]i¾i∈ {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 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 ha in his P sys em only con igu a ions in an e en s ep 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 a simila way as in he de e minis ic case. 3 Mic oscopic View: P Sys ems Modeling Molecula In e ac ions Besides modula i y and easy ex ensibili y, in a o 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 mod- eling cance ous p ocesses. Ins ead o conside ing he umou as a whole, he e we use P sys ems in o de o simula e he molecula pa hways inside he cell. 3.1 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 [17] 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. A Memb ane Compu ing View on Tumou s 109 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 objec 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 inuous 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. Acon inuous Psys em o deg ee n≥1 is a cons uc , Π= (Σ, µ, w1,. . . , wn,R,K), whe e: 1. Σ={c1, . . . , cm}is he alphabe o objec s. 2. µis a memb ane s uc u e (a oo ed ee) consis ing o nmemb anes (nodes o he ee) labeled wi h 1, . . . , n. 3. w1, . . . , wna e con inuous mul ise s associa ed wi h each memb ane o µ. 4. 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∈ {1, . . . , n}is he label o he memb ane whe e he eac ion is ca ied ou . 5. Kis a map om R×Mn×m(R+) o R+, called he a e o applica ion unc ion, 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 an ins an aneous desc ip ion o i , ha is, an assignmen o con inuous mul ise s o he memb anes o he sys em ha can be seen as an associa ion o each egion wi h he concen a ion o chemical subs ances p esen in i . Fo mally, a con igu a ion o Πis a ma ix o Mn×m(R+) whe e he elemen in ow iand column j ep esen s he mul iplici y o he objec cjin he memb ane i. Then, he a e o applica ion unc ion Kassocia es wi h each ule and each con igu a ion, a non- nega i e eal numbe conside ed as he a e o applica ion o he ule. An e olu ion Eo a con inuous P sys em associa es wi h each ins an ∈R+a con igu a ion E( ) o he sys em. 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, whe e aij( ) is he elemen in ow iand column j om E( ). So, we can desc ibe he con igu a ion 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 Π as 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). Gi en an objec cj∈Σand a memb ane i, we deno e by p oduc ioni(cj) ( esp. consump ioni(cj)) he se o ules whe e cjis a p oduc ( esp. a eac an ) in memb ane i. The e o e he eal numbe ( i( ))(cj) is de e mined by he nex o mula: ( i( ))(cj)=( i(0))(cj) + X ∈p oduc ioni(cj)Z 0 K( , E(s)) ds − −X ∈consump ioni(cj)Z 0 K( , E(s)) ds, whe e i(0) is he ini ial con inuous mul ise wi. 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,· · · , q we 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 110 M.A. Gu i´e ez-Na anjo e al. 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)). 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 . 3.2 EGFR Signalling Cascade In his subsec ion we b ie ly desc ibe a EGFR signalling cascade4. 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 isa 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 nalisa ion, ubiqui ina ion and deg ada ion. EGFR has been iden i ied as a key biological a ge o he de elopmen o no el an icance he apies. 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 phospho y 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 phospho y 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 dissoci- a 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 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. Subse- quen 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. We ha e de eloped a model o he signalling cascade using a con inuous P sys em ΠEGF = (Σ, µ, we, ws, wc,R,K). Ou model consis s in mo e ha 60 p o eins and complexes o p o eins and 160 chemical eac ions. •Alphabe : In he alphabe Σwe collec all he p o eins and complexes o p o eins ha ake pa in he signalling cascade. •Memb ane S uc u e: We conside h ee ele an egions, namely he en i onmen , he cell su ace and he cy oplasm. We ep esen hem in a nes ed memb ane s uc u e as he memb anes labeled wi h: e o he en i onmen (ex e nal egion), 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 ions ha e been ob ained om e e ences in he li e a u e. •Rules and Ra e o applica ion unc ion: In he ules we model he chemical 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 4see www.gcn.us.es/eg .pd o de ails. A Memb ane Compu ing View on Tumou s 111 ha he a e o a eac ion is p opo 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 . 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 el- e 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|EGFR( )|s 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 . 4 Final Rema ks In his pape wo app oaches b idging cance esea ch and memb ane compu ing ha e been p esen ed. A he mac oscopic le el, ac als a e nowadays one o he mos powe ul ools o desc ibing he shape o umou s in a ealis ic manne . Na u e is w i en in ac al language and we need ools o dealing easily wi h his language. In his pape we p esen a i s wo k checking whe he P sys ems p o ide an app op ia e ool o handling ac als. On he one hand, he massi e pa allelism, he synch onous applica ion o he ules, and he disc e e na u e o hei compu a ion, among o he ea u es, lead us o conside P sys ems as na u al ools o dealing wi h ac als. On he o he hand, he main d awback is ha P sys ems wo k wi h da a s uc u es which do no ha e a geome ical in ui ion, in he sense ha concep s as leng h o angle a e no in memb ane compu ing e minology. This leads us o he necessi y o gi ing a geome ical in e p e a ion o he da a o he P sys ems in o de o conside i as a ac al. A he mic oscopic le el, we show ha con inuous memb ane sys ems a e a eliable amewo k o modelling ne wo ks o biochemical signalling cascades, because he esul s ob ained using his model a e in well ag eemen wi h expe imen al da a. We in end o expand ou model o comp ise o he in e ac ions be ween p o eins which a e known o play a key ole in he egula ion o cell cycle and umou genesis. Acknowledgemen Wo k suppo ed by p ojec TIN2005-09345-C04-01 o Minis e io de Educaci´on y Ciencia o Spain, co inanced by FEDER unds, and by he P ojec o Excellence TIC-581 o he Jun a de Andaluc´ıa. Re e ences 1. J.W. Baish, R.K. Jain: F ac als and cance . Cance Resea ch, 60 (2000), 3683–3688. 2. M.F. Ba nsley: Lec u e no es on i e a ed unc ion sys ems. P oceedings o Symposia in Applied Ma he- ma ics, AMS, 39 (1989), 127–144. 3. A. B ´u, J.M. Pas o , I. Fe naud, I. B ´u, S. Melle, C. Be engue : Supe - ough dynamics on umou g ow h. Physiscal Re iew Le e s, 81, 18 (1998), 4008–4011.